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You will want to read the following for starters on understanding bitcoin cryptography.
https : en.bitcoin.it wiki Secp256k1
http : en.wikisource.org wiki NIST_Koblitz _Curves _Parameters
The naive way to break an encryption algorithm is to brute
force the key.The complexity of that attack is 2 n,where n is the key length.All cryptanalytic attacks can be viewed as shortcuts to that
method.And since the efficacy of a brute force attack is a direct function of key length,
these attacks effectively shorten the key.So if, for example,
the best attack against DES has a complexity of 239, that effectively shortens DES s 56
bit key by 17 bits.That s a really good attack,
by the way.Right now the upper practical
limit
on
brute
force
is
somewhere
under
80
bits.However, using
that
as
a guide
gives us some
indication as to how
good an attack has to be
to break any of the modern
algorithms.These days,
encryption algorithms have, at a minimum,
128
bit keys.That means any NSA cryptoanalytic breakthrough has to reduce
the effective key length by at least 48 bits in order to be practical.
http : en.wikipedia.org wiki Elliptic_Curve _DSA
http : www.reddit.com r Bitcoin comments 1 nvsn7 elliptic_curve _secp256k1 _vulnerability http : blog.ezyang.com 2011 06 the cryptography of bitcoin http : bitcoinmagazine.com 6021 bitcoin is not quantum safe and how we can fix http : en.wikipedia.org wiki Cryptographic_hash _function
http : mathworld.wolfram.com EllipticCurve.html
We can liken the finding of bitcoins to someone firing at a target,
and they are force to fire at the target while wearing a blind fold.
We can make the statistical probability of hitting the target greater if we either
1. Take off the blind fold very hard 2. Make the target bigger easier3. Create a net so that we hit the target and drag it back to us so that it' s easier to hit.
Making target bigger : Say we know what the elliptical curve that bitcoin uses,
we can can create more facets to it. We can embed different functions within it;
reduce it to other variable entirely. Then,
we have better ways of converting to potential solutions to the equation.
We need to know what the field characteristic of the bitcoin elliptical curve is.
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http : bitcoin.stackexchange.com questions 21 907 what
does the curve used in bitcoin secp256k1 look like
http : www.secg.org index.php?action secg, docs_secg
Actually secp256k1 is defined over a Galois field, not a ring of integers modulo a prime.Now,
it turns out that the secp256k1 field is a prime field and therefore isomorphic to a
ring of integers modulo a prime, but this is not true for all ECDSA curves in fact,
the "sectXXXyZ" curves for which much faster hardware exists than the "secpXXXyZ" curvescannot be described using rings of integers.See this page for an explanation of why every
finite field has a GF representation but only the prime fields have a a Z pZ representation :
en.wikipedia.org wiki Finite_field Statement
See : Fp2 via quadratic residues modulo p Not wholly true. Not necessarily correct. Here,
phenomenological velocity can be used.
Please research what is the relationship between Fp2 and bitcoin secp256k1?
http : www.reddit.com r Bitcoin comments 1 nvsn7 elliptic_curve _secp256k1 _vulnerability
"On their protocol specification wiki they say that in their scripts they provide
hexidecimal decompressed x,y coordinates
though these are really r,s values
"
So, if x and y are actually r and s, then we can boil either down to only one variable. We
can actually force r to be purely in terms of s by definition. But what are r and s ?
http : bitcoinmagazine.com 7781 satoshis genius
unexpected ways in which bitcoin dodged some cryptographic bullet Thus, elliptic curve cryptography uses an elliptic curve with two modifications.First,
the equation is now y2 x3 ax b kp, where k can be any integer and p is
some large prime number a parameter of the curve alongside a and b.Second,
x and y must be integers.Although the resulting set is hardly a curve,
surprisingly enough the same math still works,
and the restriction to integers avoids rounding errors.
y ^ 2 x ^ 3 ax b kp
In general, however, the curves fall into twocategories : pseudorandom curves and Koblitz curves.In a pseudorandom curve,
the parameters a and b are chosen by a specified algorithm essentially a hashfrom a certain seed .For secp256r1, the standard 256 bit pseudorandom curve,
the seed is c49d360886e704936a6678e1139d26b7819f7e90, giving rise to the parameters :
Vitalik Buterin : Fortunately, Bitcoin does not use pseudorandom curves;
Bitcoin uses Koblitz curves.In Bitcoin s secp256k1, the parameters are :
p 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
a 0
b 7
y ^ 2 x ^ 3 0 x 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
This is the base equation for bitcoin elliptical curves from what Vitalik says,
and Vitalik is the top notch 1 programmer in the
bizzz.
Theta BASE
y height of cone h 4 Π r2 Θ r2 Θ2
2 Π
2 Some Bitcoin for Thought.nb
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x base of cone r ^ 2 Η ^ 2 2 Π r r Θ
2 Π
y ^ 2 x ^ 3 0 x 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
4 Π r2 Θ r2 Θ2
2 Π ^ 2
2 Π r r Θ
2 Π ^ 3 0
2 Π r r Θ
2 Π 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
4 Π r2 Θ r2 Θ2
4 Π2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k 2 Π r r Θ3
8 Π3
Solve
4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3, r
r 2 4 Π2 Θ Π Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 4 213 4 Π2 Θ Π Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
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975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730
501069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995 191
503207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989
379009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329 326
252673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498 497
344752 480 k Π5 Θ4 18 336 Π4 Θ5 300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748
499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6213 3 2
13 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3, r 2 4 Π2 Θ Π Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
2 213 1 3 4 Π2 Θ Π Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
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730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6
Θ3
91 488 Π5
Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213
1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
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4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213 6 2
13 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3, r 2 4 Π2 Θ Π Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
2 213 1 3 4 Π2 Θ Π Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
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145 708 308 289 079 208 k Π3 Θ6213
1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2 6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213
6 2
13
8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
But what is zero? Is it really so nothing?
Usefulness : 0 2 Π r 2 Π x Θ r
0 2 Π r 2 Π x Θ r
So, let' s try that again :
4 Π r2 Θ r2 Θ2
2 Π ^ 2
2 Π r r Θ
2 Π ^ 3 0
2 Π r r Θ
2 Π 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
Solve4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3, r
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4 Π r2 Θ r2 Θ2
2 Π ^ 2
2 Π r r Θ
2 Π ^ 3 2 Π r 2 Π x Θ r 2 Π r r Θ
2 Π 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
4 Π r2 Θ r2 Θ2
4 Π
2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k 2 Π r r Θ3
8 Π32 Π r r Θ 2 Π r 2 Π x r Θ
2 Π
Solve 4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3
2 Π r r Θ 2 Π r 2 Π x r Θ2 Π
, r
r
2 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ23 8 Π
3
12 Π2
Θ 6 Π Θ2
Θ3
213 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 12 288 Π10 Θ 24 576 Π11 Θ
9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 6144 Π8 Θ2 27648 Π9 Θ2
30 720 Π10 Θ2 20 736 Π8x Θ2 69120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 9216 Π7 Θ3 24576 Π8 Θ3
20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 4992 Π6 Θ4 10752 Π7 Θ4
7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5
1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
144 Π4x Θ6 288 Π5
x Θ6 96 768 Π9
8 Some Bitcoin for Thought.nb
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1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9
8192 Π12 18432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12288 Π10 Θ
24576 Π11 Θ 9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27648 Π9 Θ2 30 720 Π10 Θ2 20 736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10752 Π7 Θ4 7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6
192 Π5 Θ6 128 Π6 Θ6 144 Π4 x Θ6 288 Π5 x Θ6
2
4 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22 24 2 Π4
x Π3x Θ 8 Π3 12 Π2 Θ
6 Π Θ2 Θ3313
1
3 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730
501069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995 191
503207 936 k Π8 Θ 12 288 Π10 Θ 24576 Π11 Θ
9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989
379009 920 k Π7 Θ2 6144 Π8 Θ2 27 648 Π9 Θ2 30 720 Π10 Θ2
20 736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329 326 252673 280 k Π6 Θ3 9216 Π7 Θ3 24 576 Π8 Θ3 20 480 Π9 Θ3
18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498 497
344752 480 k Π5 Θ4 4992 Π6 Θ4 10 752 Π7 Θ4 7680 Π8 Θ4
8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5 1536 Π7 Θ5
1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
Some Bitcoin for Thought.nb 9
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144 Π4x Θ6 288 Π5
x Θ6 96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12 288 Π10 Θ
24 576 Π11 Θ 9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27 648 Π9 Θ2 30 720 Π10 Θ2 20 736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24 576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10 752 Π7 Θ4 7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748
499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6
128 Π6 Θ6 144 Π4x Θ6 288 Π5
x Θ62
4 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3313,
r 2 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 3 2
23 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 12 288 Π10 Θ 24 576 Π11 Θ
9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 6144 Π8 Θ2 27648 Π9 Θ2
30 720 Π10 Θ2 20 736 Π8x Θ2 69120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 9216 Π7 Θ3 24576 Π8 Θ3
20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 4992 Π6 Θ4 10752 Π7 Θ4
7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
10 Some Bitcoin for Thought.nb
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699 468 950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5
1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
144 Π4x Θ6 288 Π5
x Θ6 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9
8192 Π12 18432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12288 Π10 Θ
24576 Π11 Θ 9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27648 Π9 Θ2 30 720 Π10 Θ2 20 736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10752 Π7 Θ4 7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6
128 Π6 Θ6 144 Π4x Θ6 288 Π5
x Θ62 4 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3313
1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 12 288 Π10 Θ 24576 Π11 Θ
9216 Π9x Θ 55296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 6144 Π8 Θ2 27 648 Π9 Θ2
30 720 Π10 Θ2 20736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 9216 Π7 Θ3 24 576 Π8 Θ3
20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 4992 Π6 Θ4 10 752 Π7 Θ4
7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
Some Bitcoin for Thought.nb 11
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300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5 1536 Π7 Θ5
1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
144 Π4x Θ6 288 Π5
x Θ6 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730 501 069 312 k Π9 8192 Π12
18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12 288 Π10 Θ
24 576 Π11 Θ 9216 Π9x Θ 55296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27 648 Π9 Θ2 30 720 Π10 Θ2 20736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24 576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46080 Π8
x Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10 752 Π7 Θ4 7680 Π8 Θ4 8064 Π6 x Θ4 17 280 Π7 x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6
128 Π6 Θ6 144 Π4x Θ6 288 Π5
x Θ62 4 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3313,
r 2 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4
x Π3
x Θ 8 Π3
12 Π2
Θ 6 Π Θ2
Θ3
3 2
23 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 12 288 Π10 Θ 24 576 Π11 Θ
9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 6144 Π8 Θ2 27648 Π9 Θ2
30 720 Π10 Θ2 20 736 Π8x Θ2 69120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
12 Some Bitcoin for Thought.nb
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326 252 673 280 k Π6 Θ3 9216 Π7 Θ3 24576 Π8 Θ3
20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 4992 Π6 Θ4 10752 Π7 Θ4
7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5
1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
144 Π4x Θ6 288 Π5
x Θ6 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9
8192 Π12 18432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12288 Π10 Θ
24576 Π11 Θ 9216 Π9x Θ 55 296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27648 Π9 Θ2 30 720 Π10 Θ2 20 736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10752 Π7 Θ4 7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6
128 Π6 Θ6 144 Π4x Θ6 288 Π5
x Θ62 4 4 8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ22
24 2 Π4
x Π3
x Θ 8 Π3
12 Π2
Θ 6 Π Θ2
Θ3
3
1
3
1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 8192 Π12 18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 12 288 Π10 Θ 24576 Π11 Θ
9216 Π9x Θ 55296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 6144 Π8 Θ2 27 648 Π9 Θ2
30 720 Π10 Θ2 20736 Π8x Θ2 69 120 Π9
x Θ2 242944 Π6 Θ3
Some Bitcoin for Thought.nb 13
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4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 9216 Π7 Θ3 24 576 Π8 Θ3
20 480 Π9 Θ3 18 432 Π7x Θ3 46 080 Π8
x Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 4992 Π6 Θ4 10 752 Π7 Θ4
7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1152 Π5 Θ5 2304 Π6 Θ5 1536 Π7 Θ5
1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6 128 Π6 Θ6
144 Π4x Θ6 288 Π5
x Θ6 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 8192 Π12
18 432 Π11x 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 12 288 Π10 Θ
24 576 Π11 Θ 9216 Π9x Θ 55296 Π10
x Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 6144 Π8 Θ2
27 648 Π9 Θ2 30 720 Π10 Θ2 20736 Π8 x Θ2 69 120 Π9 x Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 9216 Π7 Θ3
24 576 Π8 Θ3 20 480 Π9 Θ3 18 432 Π7x Θ3 46080 Π8
x Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 4992 Π6 Θ4
10 752 Π7 Θ4 7680 Π8 Θ4 8064 Π6x Θ4 17 280 Π7
x Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1152 Π5 Θ5
2304 Π6 Θ5 1536 Π7 Θ5 1728 Π5x Θ5 3456 Π6
x Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 96 Π4 Θ6 192 Π5 Θ6
128 Π6 Θ6 144 Π4x Θ6 288 Π5
x Θ6
2
4
4
8 Π4 4 Π2 Θ 8 Π3 Θ Π Θ2 2 Π2 Θ2
2
24 2 Π4x Π3
x Θ 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3313
Solve 4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3
2 Π r r Θ 2 Π r 2 Π x r Θ2 Π
, k
k 56 Π3 16 Π4r
2 8 Π3r
3 16 Π4r x 8 Π2
r2 Θ
16 Π3r
2 Θ 12 Π2r
3 Θ 8 Π3r x Θ 2 Π r
2 Θ2 4 Π2r
2 Θ2 6 Π r3 Θ2 r
3 Θ3 926 336 713 898 529 563 388 567 880 069 503 262 826 159 877 325 124 512 315 660 672 063 270 677 373 304
Π
3
14 Some Bitcoin for Thought.nb
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Solve 4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3
2 Π r r Θ 2 Π r 2 Π x r Θ2 Π
, k
k 56 Π3 16 Π4 r2 8 Π3 r3 16 Π4 r x 8 Π2 r2 Θ
16 Π3r
2 Θ 12 Π2r
3 Θ 8 Π3r x Θ 2 Π r
2 Θ2 4 Π2r
2 Θ2 6 Π r3 Θ2 r
3 Θ3 926 336 713 898 529 563 388 567 880 069 503 262 826 159 877 325 124 512 315 660 672 063 270 677 373 304
Π3c : 2.99792458 10^8
Solve
r 1 v2
c2
Θ
1v2
c2
4 Π r r Θ
2 Π r SinΒ, v
v 1. 1.12941 10
18 Θ 8.98755 1016 Θ2 3.54814 10
18SinΒ
2
12.5664 Θ Θ2 39.4784 SinΒ2,
v 1.12941 10
18 Θ 8.98755 1016 Θ2 3.54814 10
18SinΒ2
12.5664 Θ Θ2 39.4784 SinΒ2
Solve 4 Π r2 Θ r2 Θ2
4 Π 2 7
115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663 k
2 Π r r Θ3
8 Π 3
2 Π r r Θ 2 Π r 2 Π x r Θ
2 Π
, k
4 Π r2 Θ r2 Θ2
2 Π ^ 2
2 Π r r Θ
2 Π ^ 3 0
2 Π r r Θ
2 Π 7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
0 2 Π r 2 Π x Θ r 0 Θ r 2 Π r 2 Π r ^ 2 4 Π r2 Θ r2 Θ2
2 Π ^ 2 0
4 Π r2 Θ r2 Θ2
2 Π
^ 2 2 Π r r Θ
2 Π
^ 3 02 Π r r Θ
2 Π
7
k 115 792 089 237 316 195 423 570 985 008 687 907 853 269 984 665 640 564 039 457 584 007 908 834 671 663
Θ r 2 Π r 2 Π r ^ 2 4 Π r2 Θ r2 Θ2
2 Π ^ 2
Some Bitcoin for Thought.nb 15
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25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213
1
3 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730
501069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995 191
503207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989
379009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329 326
252673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498 497
344752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6
256
4 Π2 Θ Π Θ2
6
96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748
499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
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25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6213,
r 2 4 Π2 Θ Π Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
2 21
3
1 3 4 Π2
Θ Π Θ2
2
3 8 Π3
12 Π2
Θ 6 Π Θ2
Θ3
96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26
96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213
1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
18 Some Bitcoin for Thought.nb
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326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6
256
4 Π2 Θ Π Θ2
6
96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6
2
1
3
,
r 2 4 Π2 Θ Π Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
2 213 1 3 4 Π2 Θ Π Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6
Θ3
91488 Π5
Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 18336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
Some Bitcoin for Thought.nb 19
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969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6213
1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 256 4 Π2 Θ Π Θ26 96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6213,
r 2 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ2
3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
4 213 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
20 Some Bitcoin for Thought.nb
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1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 12288 Π8 Θ2 110592 Π9 Θ2
245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6 256 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ
49152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2 6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242 944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3
18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4
9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5
2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6
768 Π5 Θ6 1024 Π6 Θ6213 1
3 2 13 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331 730
501069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995 191
503207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993 989
379009 920 k Π7 Θ2 12 288 Π8 Θ2 110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329 326
252673 280 k Π6 Θ3 18 432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
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1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498 497
344752 480 k Π5 Θ4 9984 Π6 Θ4 43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12 288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6
256
16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ2
6
96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ 49 152 Π10 Θ
196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 18 432 Π7 Θ3
98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 9984 Π6 Θ4
43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748
499 699 468 950 496 k Π4 Θ5 2304 Π5 Θ5
9216 Π6 Θ5 12 288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6
768 Π5 Θ6 1024 Π6 Θ6213, r
2 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ23 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 2 2
13 1 3 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ22 3 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ 4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 12288 Π8 Θ2 110592 Π9 Θ2
245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
22 Some Bitcoin for Thought.nb
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25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6 256 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8
Θ 49152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242 944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3
18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4
9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5
2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6
1024 Π6 Θ6213 1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2 110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 18 432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 9984 Π6 Θ4 43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6 256 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ
49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
Some Bitcoin for Thought.nb 23
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6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 18 432 Π7 Θ3
98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 9984 Π6 Θ4
43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 2304 Π5 Θ5
9216 Π6 Θ5 12 288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6
768 Π5 Θ6 1024 Π6 Θ6213, r
2 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ23 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 2 2
13 1 3 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ22
3
8 Π3 12 Π2 Θ 6 Π Θ2 Θ3
96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 12288 Π8 Θ2 110592 Π9 Θ2
245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499
699 468 950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6 256 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ
49152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242 944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
24 Some Bitcoin for Thought.nb
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313 329 326 252 673 280 k Π6 Θ3
18432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4
9984 Π6 Θ4 43008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5
2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838
145 708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6
1024 Π6 Θ6213 1
6 213 8 Π3 12 Π2 Θ 6 Π Θ2 Θ3 1 3 96768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641 325 331
730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923 975 995
191 503 207 936 k Π8 Θ 49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154 969 993
989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2 110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103 313 329
326 252 673 280 k Π6 Θ3 18 432 Π7 Θ3 98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288 742 498
497 344 752 480 k Π5 Θ4 9984 Π6 Θ4 43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057 748 499 699
468950 496 k Π4 Θ5 2304 Π5 Θ5 9216 Π6 Θ5 12288 Π7 Θ5 1528 Π3 Θ6
25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145 708 308
289079 208 k Π3 Θ6 192 Π4 Θ6 768 Π5 Θ6 1024 Π6 Θ6 256 16 Π4 4 Π2 Θ 16 Π3 Θ Π Θ2 4 Π2 Θ26 96 768 Π9
1 600 709 841 616 659 085 535 445 296 760 101 638 163 604 268 017 815 157 281 461 641
325 331 730 501 069 312 k Π9 65 536 Π12 290304 Π8 Θ
4 802 129 524 849 977 256 606 335 890 280 304 914 490 812 804 053 445 471 844 384 923
975 995 191 503 207 936 k Π8 Θ
49 152 Π10 Θ 196608 Π11 Θ 362880 Π7 Θ2
6 002 661 906 062 471 570 757 919 862 850 381 143 113 516 005 066 806 839 805 481 154
969 993 989 379 009 920 k Π7 Θ2 12 288 Π8 Θ2
110592 Π9 Θ2 245760 Π10 Θ2 242944 Π6 Θ3
4 001 774 604 041 647 713 838 613 241 900 254 095 409 010 670 044 537 893 203 654 103
313 329 326 252 673 280 k Π6 Θ3 18 432 Π7 Θ3
98 304 Π8 Θ3 163840 Π9 Θ3 91 488 Π5 Θ4
1 500 665 476 515 617 892 689 479 965 712 595 285 778 379 001 266 701 709 951 370 288
742 498 497 344 752 480 k Π5 Θ4 9984 Π6 Θ4
43 008 Π7 Θ4 61 440 Π8 Θ4 18 336 Π4 Θ5
300 133 095 303 123 578 537 895 993 142 519 057 155 675 800 253 340 341 990 274 057
748 499 699 468 950 496 k Π4 Θ5 2304 Π5 Θ5
9216 Π6 Θ5 12 288 Π7 Θ5 1528 Π3 Θ6
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25011091 275 260 298 211 491 332 761 876 588 096 306 316 687 778 361 832 522 838 145
708 308 289 079 208 k Π3 Θ6 192 Π4 Θ6
768 Π5 Θ6 1024 Π6 Θ6213
Solve
2 Π 2 Π r 2 Π r2 Η2
Θ2
4 Π r2 2 r2 Θ
2 4 Π r2 Θ r2 Θ2
, r
0
2 Π 2 Π r 2 Π r2 Η2
Θ2
4 Π r2 2 r2 Θ
2 4 Π r2 Θ r2 Θ2
0
Solve2 Π 2 Π r 2 Π r2 Η2
Θ2
4 Π r2 2 r2 Θ
2 4 Π r2 Θ r2 Θ2
Θ r
2 Π r 2 Π
r ^ 2 Η ^ 2
1
4 Π Θ Θ2
2 Π SinΒ , r
Solve2 Π 2 Π r 2 Π r2 Η2
Θ2
4 Π r2 2 r2 Θ
2 4 Π r2 Θ r2 Θ2
Θ r 2 Π r 2 Π r ^ 2 4 Π r2 Θ r2 Θ2
2 Π ^ 2 1
4 Π Θ Θ2
2 Π SinΒ , Η
Η
1
32 Π3 512 Π4
r Θ2 1024 Π5r
2 Θ2 512 Π4r r
2 2 Π Θ2Θ2 64 Π3
r2 Θ3 512 Π4
r2 Θ3 64 Π2 Θ4
256 Π3r Θ4 48 Π2
r2 Θ4 512 Π4
r2 Θ4 128 Π2
r2 2 Π Θ2
Θ4 256 Π3r r
2 2 Π Θ2Θ4
128 Π2r Θ5 4 Π r
2 Θ5 512 Π3r
2 Θ5 128 Π2r r
2 2 Π Θ2Θ5 r
2 Θ6
128 Π2r
2 Θ6 r2 Θ7
4 Π Θ 256 Π4
r Θ r2 4 Π Θ Θ 64 Π3
r Θ2r
2 4 Π Θ Θ
26 Some Bitcoin for Thought.nb
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256 Π3r Θ2 4 Π Θ Θ CscΒ 64 Π Θ4 4 Π Θ Θ CscΒ 128 Π2
r Θ4 4 Π Θ Θ CscΒ
64 Π r2 2 Π Θ2
Θ4 4 Π Θ Θ CscΒ 64 Π r Θ5 4 Π Θ Θ CscΒ
32 Π Θ3 4 Π Θ Θ r2 4 Π Θ Θ CscΒ 8 Θ4 4 Π Θ Θ r
2 4 Π Θ Θ CscΒ
8 Θ5
4 Π Θ
Θ r2
4 Π Θ
Θ Csc
Β
4 Π Θ 64 Π Θ5 CscΒ
2 16 Θ6 CscΒ2
Solve1 4 Π Θ Θ2
2 Π SinΒ Θ r 2 Π r 2 Π r ^ 2 4 Π r2 Θ r2 Θ2
2 Π ^ 2 , r
Solve1 2 Π SinΒ
4 Π Θ Θ2
Θ r 2 Π r 2 Π r ^ 2 4 Π r2 Θ r2 Θ2
2 Π ^ 2 , r
28 Some Bitcoin for Thought.nb
