Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - PERFECT NUMBERS AND MERSENNE'S PRIME

7/30/2019 Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - PERFECT NUMBERS AND MERSENNE'S PRIME

1/74

STUDY ON THE PERFECT NUMBERS AND MERSENNE'S PRIME WITH

NEW DEVELOPMENTS.

POSSIBLE MATHEMATICAL CONNECTIONS WITH SOME SECTORS OF

STRING THEORY

Pier Francesco Roggero, Michele Nardelli1,2

, Francesco Di Noto

1 Dipartimento di Scienze della Terra

Universit degli Studi di Napoli Federico II, Largo S. Marcellino, 10

80138 Napoli, Italy

2 Dipartimento di Matematica ed Applicazioni R. Caccioppoli

Universit degli Studi di Napoli Federico II Polo delle Scienze e delle Tecnologie

Monte S. Angelo, Via Cintia (Fuorigrotta), 80126 Napoli, Italy

Abstract

In this paper we show that Perfect Numbers are only even plus many other interesting relations

about Mersennes prime. Furthermore, we describe also various equations, lemmas and theoremsconcerning the expression of a number as a sum of primes and the primitive divisors of Mersenne

numbers. In conclusion, we show some possible mathematical connections between some equations

regarding the arguments above mentioned and some sectors of string theory (p-adic and adelic strings

and Ramanujan modular equation linked to the modes corresponding to the physical vibrations of the

bosonic strings).


2/74

Versione 1.0

14/12/2012

Pagina 2 di 74

2

Index:

1 PERFECT NUMBERS ......................................................................................................................... 32 NUMBERS OF FORM

p2 - 2: PREDICTION AND FREQUENCY 'OF PRIME NUMBERS ......... 82.1 SUMMARY: ........................................................................................................................24

3. CONNECTION NUMBERSp2 - 2 WITH MERSENNE PRIMES .................................................25

3.1PREDICTIONMERSENNENUMBERS............................................................................263.2 FACTORIZATION OF 5212 - 2 (A NUMBER OF 157 DIGITS).......................................293.3 PRIME NUMBERS ARISING FROM OTHER FORMULAS ..........................................313.4 PREDICTION OF THE FACTORS P WHICH GIVE RISE TO PRIME NUMBERS

WITHIN A GROUP pa b ......................................................................................................42

4. ON SOME EQUATIONS, LEMMAS AND THEOREMS CONCERNING THE EXPRESSION OF

A NUMBER AS A SUM OF PRIMES...48

4.1 ON SOME THEOREMS AND EQUATIONS CONCERNING THE PRIMITIVE

DIVISORS OF MERSENNE NUMBERS.61

5. MATHEMATICAL CONNECTIONS WITH P-ADIC AND ADELIC STRINGS AND WITHRAMANUJAN MODULAR EQUATION (APPROXIMATION TO ).65

5.1 MATHEMATICAL OBSERVATIONS OF FRANCESCO DI NOTO..70


3/74

Versione 1.0

14/12/2012

Pagina 3 di 74

3

1 PERFECT NUMBERS

We show that the perfect numbers are just even.

Consider any integer positive number n and we see to divide it by all the divisors of the powers of 2

summing all its proper divisors, i.e. except for n:

1 + 2 +2n + 4 +

4n + 8 +

8n + 16 +

16n + 32 +

32n + = n

If we divide n by 2 we add, obviously,2

nand so on.

The sum of all the powers of 2 gives, with p elements:

p2 - 1

While leading to the second member all the (p-1) factors in n we have:

n ( 2

n

+ 4

n

+ 8

n

+ 16

n

+ 32

n

+) = 12

n

p

then:

1 + 2 + 4 + 8 + 16 + 32 + = n (2

n+

4

n+

8

n+

16

n+

32

n+)


4/74

Versione 1.0

14/12/2012

Pagina 4 di 74

4

p2 - 1 = 12

np

And this is proved the formula that a perfect number is calculated by the following equation:

12 p (p2 - 1) = n

with p prime.

p must be prime becausep2 - 1 should give a prime number.

In fact, multiplying p2 - 1 * 12 p we must not have other divisors except the powers of 2, or, p2 - 1

and the prime numberp2 - 1.

We see some examples:

1) If we choose p = 3 (p prime) we have:

1 + 2 +4 = n ( 2

n

+ 4

n

)

7 =4

n

32 - 1 = 132

n


5/74

Versione 1.0

14/12/2012

Pagina 5 di 74

5

And we get the perfect number n = 28

----------------------------------------------------------------------------------------------------------------------------

2) If we choose p = 4 (p is not prime) we have:

1 + 2 + 4 + 8= n (2

n+

4

n+

8

n)

15 =8

n

42 - 1 = 142

n

n = 120 abundant number.

----------------------------------------------------------------------------------------------------------------------------

If we now repeat the algorithm with any integer positive number n and we see to divide it for all

divisors of powers of 3 by summing all its proper divisors, i.e. except for n, we have:

1 + 3 + 9 + 27 + 81 + = n (3

n+

9

n+

27

n+

81

n+ +)

2

13p = 1

3-p2-p1-p

3

n....)33(3

p


6/74

Versione 1.0

14/12/2012

Pagina 6 di 74

6

Now it is IMPOSSIBLE that the numerator of the 2nd member goes to 1.

More precisely it is not possible to have an integer number but only a fractional number, because at

numerator we have a number divisible by 3, while at denominator we have a prime number different

from 3.

For example, for p = 3:

1 + 3 + 9 = n (3

n+

9

n)

13 =9

5n

2

133 = 13

2-31-3

3

n1)3(3

n =5

117


7/74

Versione 1.0

14/12/2012

Pagina 7 di 74

7

If we now repeat the algorithm with any integer positive number n and we see to divide it for all the

divisors of the powers of another prime number (5, 7, ...) summing all its proper divisors, i.e. except

for the same n we ALWAYS have:

It is IMPOSSIBLE that the numerator of the 2nd member goes to 1.


numerator we have a number divisible by the prime number p that we have chosen, while at

denominator we have a prime number different from p ( p).

If we choose any integer positive number n and we see to divide it for all the divisors of the powers of

another composite number (6, 9, ...) summing all its proper divisors, i.e. except for the same n we

have ALWAYS and a FORTIORI (because between the factors we have also the factors of the

composite number) that:

It is IMPOSSIBLE that the numerator of the 2nd member goes to 1.


numerator we have a number divisible by the composite number k that we have chosen, while at

denominator we have a prime number.

This shows that the only perfect numbers are of the form:

12 p (p2 - 1) = n

with p prime and that can be only EVEN.

These are therefore perfect numbers that can only be EVEN, the odds cannot be there for what that we

have seen before and then it comes to seeking out the Mersenne primes to find a perfect number.


8/74

Versione 1.0

14/12/2012

Pagina 8 di 74

8

2 NUMBERS OF FORM p2 - 2: PREDICTION AND FREQUENCY OF PRIME

NUMBERS

We see the characteristic of the numbers:

p2 - 2

with p positive integer with p> = 2

Tab 1:

p 2^p 2 Factorizationof 2^p 2

2 2 23 6 2 34 14 2 75 30 2 3 56 62 2 317 126 2 3^278 254 2 1279 510 2 3 5 17

10 1022 2 7 7311 2046 2 3 11 3112 4094 2 23 8913 8190 2 3^2571314 16382 2 819115 32766 2 3 43 12716 65534 2 7 31 15117 131070 2 3 5 17 25718 262142 2 13107119 524286 2 3^3 7 19 7320 1048574 2 52428721 2097150 2 3 5^211 31 4122 4194302 2 7^2127 33723 8388606 2 3 23 89 68324 16777214 2 47 178481


9/74

Versione 1.0

14/12/2012

Pagina 9 di 74

9

25 33554430 2 3^25 7 13 17 24126 67108862 2 31 601 180127 134217726 2 3 2731 819128 268435454 2 7 73 26265729 536870910 2 3 5 29 43 113 12730 1073741822 2 233 1103 208931 2147483646 2 3^27 11 31 151 33132 4294967294 2 214748364733 8589934590 2 3 5 17 257 6553734 17179869182 2 7 23 89 59947935 34359738366 2 3 43691 13107136 68719476734 2 31 71 127 12292137 137438953470 2 3^35 7 13 19 37 73 10938 274877906942 2 223 61631817739 549755813886 2 3 174763 52428740 1099511627774 2 7 79 8191 12136941 2199023255550 2 3 5^211 17 31 41 6168142 4398046511102 2 13367 16451135343 8796093022206 2 3^27^243 127 337 541944 17592186044414 2 431 9719 209986345 35184372088830 2 3 5 23 89 397 683 211346 70368744177662 2 7 31 73 151 631 2331147 140737488355326 2 3 47178481 279620348 281474976710654 2 2351 4513 1326452949 562949953421310 2 3^25 7 13 17 97 241 257 67350 1,1259E+15 2127443267679859351 2,2518E+15 2311312516011801405152 4,5036E+15 2 7103 2143 1111913107153 9,0072E+15 2355315716132731819154 1,80144E+16 26361694312039440155 3,60288E+16 23^4719738721126265756 7,20576E+16 2233189881 3191 20196157 1,44115E+17 2351729431131271579032158 2,8823E+17 2732377 524287121284759 5,76461E+17 235923311032089303316960 1,15292E+18 2179951320343178033761 2,30584E+18 23^25^271113314161151331132162 4,61169E+18 2230584300921369395163 9,22337E+18 23715827883214748364764 1,84467E+19 27^27312733792737649657


10/74

Versione 1.0

14/12/2012

Pagina 10 di 74

10

65 3,68935E+19 2351725764165537670041766 7,3787E+19 231819114529514355811167 1,47574E+20 23^272367896832085759947968 2,95148E+20 219370772176183825728769 5,90296E+20 235137953263174369113107170 1,18059E+21 27471784811005267893803971 2,36118E+21 23113143711272818617112292172 4,72237E+21 22284794854412121288583373 9,44473E+21 23^35713171937731092414333873774 1,88895E+22 24392298041936197313275 3,77789E+22 2322317772578108361631817776 7,55579E+22 273115160118011008011056720177 1,51116E+23 23522945717476352428752531378 3,02231E+23 2238912758128364324911295979 6,04463E+23 23^2779273181911213692236689180 1,20893E+24 22687202029703111349113976781 2,41785E+24 235^21117314125761681427825536182 4,8357E+24 27732593711192626579768583983 9,67141E+24 238313367164511353883141869784 1,93428E+25 216757912614113275649087721 85 3,86856E+25 23^257^2132943113127337142954191444986 7,73713E+25 231131071952097280633375843187 1,54743E+26 2343197192099863293203100740388 3,09485E+26 27233110320894177985773715546389 6,1897E+26 23517238935339768321132931542412790 1,23794E+27 2618970019642690137449562111 91 2,47588E+27 23^3711193173151331631233111883700192 4,95176E+27 212791181911129011532314047153793 9,90352E+27 235472771013165730269178481279620394 1,9807E+28 27214748364765881228865355307995 3,96141E+28 23283235145131326452916576853752196 7,92282E+28 2311915242874207787513032715267197 1,58456E+29 23^257131797193241257673655372225337798 3,16913E+29 21144713842607235828485645766393 99 6,33825E+29 234312743639531272974432676798593

100 1,26765E+30 2723738919915364959974933057806959101 2,5353E+30 235^3113141101251601180140518101268501102 5,0706E+30 27432339208719341117531003194129103 1,01412E+31 23^271033072143285765291111943691131071104 2,02824E+31 225501837993976656429941438590393


11/74

Versione 1.0

14/12/2012

Pagina 11 di 74

11

105 4,05648E+31 2351753157161327318191858001308761441106 8,11296E+31 27^2317112715133729191106681122921152041107 1,62259E+32 231076361694312039440128059810762433108 3,24519E+32 2162259276829213363391578010288127 109 6,49037E+32 23^4571319377310987211246241262657279073110 1,29807E+33 2745988807870035986098720987332873111 2,59615E+33 2311^22331896838812971319120196148912491112 5,1923E+33 2722332167926295457319020217616318177113 1,03846E+34 23517294311312725751531579032154410972897114 2,07692E+34 23391232796599318685691066818132868207115 4,15384E+34 23^27571323771747635242871212847160465489116 8,30767E+34 231471495117848140369612646507710984041117 1,66153E+35 23559233110320893033169107367629 536903681118 3,32307E+35 27737993765538191861131213697830118297119 6,64614E+35 2328333717117995118247260413203431780337120 1,32923E+36 21272392023113107162983048367131105292137121 2,65846E+36 23^25^271113173141611512413311321616814562284561122 5,31691E+36 223897271786393878363164227858270210279 123 1,06338E+37 23 7686143364045646512305843009213693951124 2,12676E+37 27133673887047164511353177722253954175633125 4,25353E+37 23555818681494773847737158278832147483647126 8,50706E+37 23160118012690898060014710883168879506001127 1,70141E+38 23^37^219437312733754199273764965777158673929128 3,40282E+38 2170141183460469231731687303715884105727 129 6,80565E+38 2351725764165537274177670041767280421310721130 1,36113E+39 274319719209968311053036065049294753459639 131 2,72226E+39 23113113127314098917623851145295143558111132 5,44452E+39 226310350794431055162386718619237468234569 133 1,0889E+40 23^257132367893976832113208573127095994794327489134 2,17781E+40 2127524287163537220852725398851434325720959 135 4,35561E+40 2373276571937077217618382572876713103182899136 8,71123E+40 2731731512716312331126265734803149971617830801137 1,74225E+41 23517^213795326317436911310713546892879347902817


12/74


13/74

Versione 1.0

14/12/2012

Pagina 13 di 74

13

167,57912614113275649087721 83k+84191,420778751,30327152671 95k+96

193,22253377 96k+97199,153649,599749,33057806959 99k+100

223,616318177 37k+38229,525313 76k+77

233,1103,2089 29k+30239,20231,62983048367,131105292137 119k+120

17,241 24k+25251,4051 50k+51

17,257 16k+17263,10350794431055162386718619237468234569 131k+132

271,348031,49971617830801 135k+136277,1013,1657,30269 92k+93

281,86171 70k+71283,165768537521 94k+95

307,2857,6529 102k+10331,331 30k+31

127,337 21k+22353,29315424127 88k+89

397,2113 44k+45431,9719,2099863 43k+44

433,38737 72k+73439,2298041,9361973132 73k+74

571,160465489 114k+115577,487824887233 144k+145

601,1801 25k+26631,23311 45k+46

641,6700417 64k+6589,683 22k+23

881,3191,201961 55k+56911,112901153,23140471537 91k+92937,6553,86113,7830118297 117k+118

1429,14449 84k+851777,25781083 74k+75

2351,4513,13264529 47k+482593,71119, 97685839 81k+82

2687,202029703,1113491139767 79k+802731,8191 26k+27


14/74

Versione 1.0

14/12/2012

Pagina 14 di 74

14

2833,37171,1824726041 118k+1193391,23279,65993,1868569,1066818132868207 113k+114

4177,9857737155463 87k+885153,54410972897 112k+113

127,5419 42k+436361,69431,20394401 53k+54

8191 13k+1411447,13842607235828485645766393 97k+98

14951,4036961,2646507710984041 115k+11613367,164511353 41k+42

29191,10668,152041 105k+10632377,1212847 57k+5843691,131071 34k+35

31,61681 40k+4117,65537 32k+33

87211,262657 54k+5592737,649657 63k+64

100801,10567201 75k+76131071 17k+18

174763,524287 38k+39179951,3203431780337 59k+60

228479,48544121,212885833 71k+727,262657 27k+28

32167926295457319020217 111k+112524287 19k+20

89,599479 33k+34724153,158822951431,5782172113400990737 143k+144

858001,308761441 104k+105178481,2796203 46k+47246241, 279073 108k+109

127,15790321 56k+5731,18837001 90k+917,22366891 78k+79

48912491 110k+111107367629,536903681 116k+117

715827883,2147483647 62k+63745988807,870035986098720987332873 109k+110

193707721,761838257287 67k+682147483647 31k+32

2550183799,3976656429941438590393 103k+104


15/74

Versione 1.0

14/12/2012

Pagina 15 di 74

15

31,4278255361 80k+814375578271,646675035253258729 141k+142

61,4562284561 120k+1212099863,2932031007403 86k+87

127,4363953127297 98k+99127,4432676798593 49k+50

2690898060014710883168879506001 125k+1267432339208719,341117531003194129 101k+102

7,10052678938039 69k+7031,145295143558111 65k+66

127,581283643249112959 77k+787,658812288653553079 93k+94,2305843009213693951 61k+62

31,9520972806333758431 85k+867,11053036065049294753459639 129k+130,618970019642690137449562111 89k+90

,162259276829213363391578010288127 107k+108127,163537220852725398851434325720959 133k+13431,2679895157783862814690027494144991 145k+146

,170141183460469231731687303715884105727 127k+128


16/74

Versione 1.0

14/12/2012

Pagina 16 di 74

16

TAB 2:

factors frequency

2 always3 2k+17 3k+4

3,5 4k+531 5k+6

127 7k+83,5,17 8k+9

7,73 9k+1011,31 10k+1123,89 11k+12

5,7,13 12k+138191 13k+14

43,127 14k+1531,151 15k+1617,257 16k+17131071 17k+18

7,19,73 18k+19524287 19k+2031,41 20k+21

127,337 21k+2289,683 22k+23

47,178481 23k+2417,241 24k+25

601,1801 25k+262731,8191 26k+277,262657 27k+28

3,5,43,29,113,127 28k+29233,1103,2089 29k+30

31,331 30k+312147483647 31k+32

17,65537 32k+3389,599479 33k+34

43691,131071 34k+3571,122921 35k+36

37,109 36k+37


17/74

Versione 1.0

14/12/2012

Pagina 17 di 74

17

223,616318177 37k+38174763,524287 38k+39

79,121369 39k+4031,61681 40k+41

13367,164511353 41k+42127,5419 42k+43

431,9719,2099863 43k+44397,2113 44k+45

631,23311 45k+46178481,2796203 46k+47

2351,4513,13264529 47k+4897,673 48k+49

127,4432676798593 49k+50251,4051 50k+51

103,2143,11119 51k+5253,157,1613 52k+53

6361,69431,20394401 53k+5487211,262657 54k+55

881,3191,201961 55k+56127,15790321 56k+57

32377,1212847 57k+5859,3033169 58k+59

179951,3203431780337 59k+6061,1321 60k+61

,2305843009213693951 61k+62715827883,2147483647 62k+63

92737,649657 63k+64641,6700417 64k+65

31,145295143558111 65k+6667,20857 66k+67

193707721,761838257287 67k+68137,953,26317 68k+69

7,10052678938039 69k+70281,86171 70k+71

228479,48544121,212885833 71k+72433,38737 72k+73

439,2298041,9361973132 73k+741777,25781083 74k+75

100801,10567201 75k+76229,525313 76k+77


18/74

Versione 1.0

14/12/2012

Pagina 18 di 74

18

127,581283643249112959 77k+787,22366891 78k+79

2687,202029703,1113491139767 79k+8031,4278255361 80k+81

2593,71119, 97685839 81k+8283 82k+83

167,57912614113275649087721 83k+841429,14449 84k+85

31,9520972806333758431 85k+862099863,2932031007403 86k+87

4177,9857737155463 87k+88353,29315424127 88k+89

,618970019642690137449562111 89k+9031,18837001 90k+91

911,112901153,23140471537 91k+92277,1013,1657,30269 92k+93

7,658812288653553079 93k+94283,165768537521 94k+95

191,420778751,30327152671 95k+96193,22253377 96k+97

11447,13842607235828485645766393 97k+98127,4363953127297 98k+99

199,153649,599749,33057806959 99k+100101, 8101,268501 100k+101

7432339208719,341117531003194129 101k+102307,2857,6529 102k+103

2550183799,3976656429941438590393 103k+104858001,308761441 104k+105

29191,10668,152041 105k+106107,28059810762433 106k+107

,162259276829213363391578010288127 107k+108246241, 279073 108k+109

745988807,870035986098720987332873 109k+11048912491 110k+111

32167926295457319020217 111k+1125153,54410972897 112k+113

3391,23279,65993,1868569,1066818132868207 113k+114571,160465489 114k+115

14951,4036961,2646507710984041 115k+116107367629,536903681 116k+117


19/74

Versione 1.0

14/12/2012

Pagina 19 di 74

19

937,6553,86113,7830118297 117k+1182833,37171,1824726041 118k+119

239,20231,62983048367,131105292137 119k+12061,4562284561 120k+121

2690898060014710883168879506001 125k+126,170141183460469231731687303715884105727 127k+128

7,11053036065049294753459639 129k+130131,409891,7623851 130k+131

263,10350794431055162386718619237468234569 131k+132127,163537220852725398851434325720959 133k+134

271,348031,49971617830801 135k+136139,168749965921 138k+139

4375578271,646675035253258729 141k+142724153,158822951431,5782172113400990737 143k+144

577,487824887233 144k+14531,2679895157783862814690027494144991 145k+146

149,593,184481113,231769777 148k+149


20/74

Versione 1.0

14/12/2012

Pagina 20 di 74

20

These numbers have the particularity that we can "predict" the prime factors of which they are

composed and we thus obtain a their fast factorization.

In fact there is a FREQUENCY (periodicity) of the individual prime factors.

These numbers end all with even digits 0, 2, 4 and 6 (8 is excluded).

The cadence, to be precise, is always 2, 6, 4, 0.

These are all divisible, obviously, by 2.

Are also divisible by 3 all the numbers with p that is odd because:

p2 - 2 = 2 (1-p2 - 1) = 2 (

2n2 - 1)

We can consider the even exponent 2n, because with p that is odd, p - 1, gives an even exponent:

(2n2 - 1) = k

a3

If a = 1:2n2 = 3k + 1 always verified. For any n, we have k, and therefore always divisible by 3.

If a = 2:2n2 = 9k + 1 verified only for n = 3h. For h = 1, 2, 3 ..., we have k that satisfies the precedent

equation.

So are divisible by 3 when the exponent p satisfies this equation:

2k + 1 = p

and thence when p is equal to:

p = 3, 5, 7, 9,...

k is a positive integer k> = 1.

Are therefore divisible by 3 all the numbers with p that is odd


21/74

Versione 1.0

14/12/2012

Pagina 21 di 74

21

--------------------------------------------------------------------------------------------------------------------------

Instead, the numbers are divisible by 5 when the exponent p satisfies this equation:

4k +5 = p


p = 5, 9, 13, 17,...


On the other hand the numberp2 - 2 ends with 0.

--------------------------------------------------------------------------------------------------------------------------

Instead, the numbers are divisible by 7 when the exponent p satisfies this equation:

3k +4 = p


p = 4, 7, 10, 13,...


--------------------------------------------------------------------------------------------------------------------------


22/74

Versione 1.0

14/12/2012

Pagina 22 di 74

22

Always divisible by p prime:

Since we can consider the even exponent 2n, because p is odd and prime, p - 1 gives an even exponent

and we have:

(2n2 - 1) = k (2n+1) always verified by (2n +1) prime, we have an integer k that satisfies the equation,

and thence are always divisible by (2n +1) for any n and therefore for any p prime.

p - 1 = 2n

p = 2n +1

Thencep2 - 2 is always divisible by p, which is a prime number.

We can then "PREDICT" the prime factors of an arbitrary big numberp2 - 2.

If then we choose p prime we know that is already divisible by p, as well as for 2 and 3 (because p is

also odd).

This can be seen from the fact that:

(2n2 - 1) = 1 + 2 + 4 + 8 + 16 + . +

1-2n2

If there are no prime factors already known a priori we have NEW PRIME NUMBERS VERY

LARGE THAT HAVE ALSO THEIR A FREQUENCY INSIDE THE GROUP OF NUMBERSp2 - 2

Examples:

If we want to factorize1212 - 2 = 2.658.455.991.569.831.745.807.614.120.560.689.150 (a number of

37 digits) we have:

2 3^2 5^2 7 11 13 17 31 41 61 151 241 331 1321 61681 4562284561


23/74

Versione 1.0

14/12/2012

Pagina 23 di 74

23

Obviously it's divisible by 2, 3 (121 is not good because it isnt prime number).

Looking at the Table 2 we know ALREADY A PRIORI that is divisible by all the factors 2 3^2 5^2 7

11 13 17 31 41 61 151 241 331 1321 61681 and 4.562.284.561

Divisible by 3 because p = 121 odd (2k + 1 = 121; k = 60)

Divisible by 5 because 4k +5 = 121, k = 29


Divisible by 11 because 10k +11 = 121, k = 11Divisible by 13 because 12k +13 = 121, k = 9

Divisible by 17 because 8k + 9 = 121, k = 14



Divisible by 61 and 1321 because 60k +61 = 121, k = 1



Divisible by 331 because 30k +31 = 121; k = 3

Divisible by 61 681 because 40k +41 = 121; k = 2

Divisible by 4,562,284,561 because 120k +121 = 121; k = 0


24/74

Versione 1.0

14/12/2012

Pagina 24 di 74

24

2.1 SUMMARY:

With these numbers therefore it is possible:

1) Factorize quickly because we know already a priori certain factors of the numberp2 - 2.

2) There are, obviously, also all the prime numbers and they have all a their FREQUENCY, simply

becausep2 - 2 is divisible by p when p is prime.

3) Find large prime numbers arbitrary and that have in their prime factors also their frequency.


25/74

Versione 1.0

14/12/2012

Pagina 25 di 74

25

3. CONNECTION NUMBERS p2 - 2 WITH MERSENNE PRIMES

p2 - 2 = 2 (1-p2 - 1)

Consequently, the prime numbersp2 - 1 are simply "shifted of one" with respect to the numbers

p2 -

2.

The prime factors that give rise to the Marsenne primes Mp of the groupp

2 - 2 are only two: 2 andMp.

For example, for p = 13 we have:

132 1 = 8191

142 2 = 16382 = 2*8191

We may use the Mersenne primes (p2 - 1) and we obtain prime numbers A, B, ...Z larger then the

MAX Mersenne prime yet known:

2 ^ (p2 1) -2 = 2*3* * (

p2 1) * A*B* Z

Then choosing the largest Mersenne prime known, we have:

(p2 1) = ( 431126092 - 1)

and thence its factorization:

12431126092 -2 = 2*3* * ( 431126092 - 1) * A*B* Z

It is therefore to perform a gigantic division to find prime numbers larger than those of Mersenne.

The frequency of Mersenne prime, inside of (p2 - 2) is then given by the following equation:

( 431126092 - 2) * k + ( 431126092 - 1) = p


26/74

Versione 1.0

14/12/2012

Pagina 26 di 74

26

3.1PREDICTIONMERSENNENUMBERS

If we look at the Mersenne numbers highlighted in blue in Table 1b with their frequency in the groupsp2 - 2, or

p2 - 1 which is equivalent because the prime factors are simply shifted by 1, we can

observe:

To be a Mersenne number, the prime number p must obey the following rule:

1) When the first number is unique is a Mersenne prime Mp.They are, of course, connected with the

prime p.

2) When we have a pair of prime numbers only one of the two may be a Mersenne prime number

(except for the pair 3, 5 but the 3 is the default for p odd).

So if there is already a known prime number Mersenne the other cannot be a Mp.

If there isnt already a known prime number Mersenne it isnt said that one of the two can be.

3) When we have 3 or more factors may happen that only one of the new factors give an Mp based onthe decomposition of the associated prime number and see the TAB. 1b


27/74

Versione 1.0

14/12/2012

Pagina 27 di 74

27

We see cases with a pair of prime numbers with given a certain frequency.

11, 31 10k+11

p = 11 cannot give an Mp because it is accompanied by 31 which gives M31.

------------------------------------------------------------------------------------------------------------------------

23, 89 11k+12

12 4094 2 23 89

In the decomposition of p = 12 there is the prime 89 which gives M89, thence 23 cannot be.

------------------------------------------------------------------------------------------------------------------------

43, 127 14k+15

p = 43 cannot give an Mp because it is accompanied by 127 which gives M127.

------------------------------------------------------------------------------------------------------------------------

61, 1321 60k+61

61 2,30584E+18 2 3^2 5^2 7 11 13 31 41 61 151 331 1321

From the decomposition we have many factors (7, 31 exist).

One of the two, 61 and 1321, could be a Mersenne number. In fact, the 61 is a Mersenne prime.


28/74

Versione 1.0

14/12/2012

Pagina 28 di 74

28

83, 8831418697 82k+83

83 9,67141E+24 2 3 83 13367 164511353 8831418697

We already know that 83 is not a Mersenne prime.

The 83 is accompanied by 13367, 164511353, 8831418697

13,367 164,511,353 and their frequency 41k + 42. 13367 we know that it is not, but 164,511,353

might be a Mersenne prime.

Remains 8,831,418,697

p = 8.831.418.697 can be A NEW prime that give an Mp:

(As in the decomposition of p = 83 there isnt Mersenne primes, except of course that there is always

3, p = 8831418697 and new Mersenne prime)

M = 6978.831.418.2 - 1

107, 28059810762433 106k+107

107 1,62259E+32 2 3 107 6361 69431 20394401 28059810762433

The factor28059810762433

can not give an Mp because it is accompanied from 107 that give M107In the decomposition of p = 107 there arent Mersenne numbers (remember that a priori we do not

know yet!), so 107 is a Mersenne prime.


29/74

Versione 1.0

14/12/2012

Pagina 29 di 74

29

3.2FACTORIZATIONOF 5212 - 2 (A NUMBER OF 157 DIGITS)

The procedure is the following:

Before we factorizes 520

520 = 2^3 5 13

We can find all its factors and connect to the TAB 2:

Factors of

520 Factors of di 5212 - 2

2 3

4 5

8 17

5 31

13 8191

10 11

20 41

40 61681

26 2731

52 53, 157, 1613

104 858001, 308761441

65 145295143558111

130 131, 409891, 7623851

So we already know the following factors, including the 521 because it is a prime number:

520 = 2^3 5 13

5212 - 2 = 2 3 5 11 17 31 41 53 131 157 521 1613 2731 8191 61681 409891 858001 7623851

308761441 145295143558111,

Surely 521 is a single prime number of TAB. 2, i.e. the first time that it appears is only factorizing 5212

- 2.

The Mersenne prime comes from the factorization of 5222 - 2 = 2 M521The TABLES 1b and 2, thence, have a new record with:


30/74

Versione 1.0

14/12/2012

Pagina 30 di 74

30

521, another factors. 520k+521


31/74

Versione 1.0

14/12/2012

Pagina 31 di 74

31

3.3 PRIME NUMBERS ARISING FROM OTHER FORMULAS

From the above considerations there are infinitely many other groups such as p2 - 1 as those of

Mersenne.

For example, inside p3 - 2 (see TAB. 5a,b) and p1999 - 2 (see TAB. 6a,b), p12 - 1 (TAB 7a,b), p12 -

5 (TAB 8a,b) and p12 + 5 (TAB 9a,b) have a periodicity of their factors and we can compile lists of

prime numbers.

With the group p3 - 2 there are prime numbers with:

p = 2, 4, 5, 6, 9, 22,

Let's see the TAB 5a:

p 3^p -2 Factorization of 3^p -22 7 7

3 25 5^2

4 79 79

5 241 241

6 727 727

7 2185 5 19 23

8 6559 7 937

9 19681 19681

10 59047 137 431

11 177145 5 71 49912 531439 113 4703

13 1594321 197 8093

14 4782967 7 17 40193

15 14348905 5 2869781

16 43046719 89 483671

17 129140161 29 47 94747

18 387420487 23 3617 4657

19 1162261465 5 232452293

20 3486784399 7 498112057


32/74

Versione 1.0

14/12/2012

Pagina 32 di 74

32

21 10460353201 3719 281267922 31381059607 31381059607

23 94143178825 5^2 3765727153

24 282429536479 31 6679 1364071

25 847288609441 19 44594137339

26 2541865828327 7^2 401 1033 125231

27 7625597484985 5 43 2693 13170403

28 22876792454959 4273 5353801183

29 68630377364881 23 101 7103 4159349

30 205891132094647 17 263 46050353857

31 617673396283945 5 419491 294487079

TAB 5b:

p prime frequency

5 4k+3

7 6k+2

17 16k+14

19 18k+723 11k+7

Group numbers p3 - 2 all end with the numbers 7, 5, 9 and 1 (the series has this cyclical).

All group numbers p3 - 2 have sum of their digits 5.

Therefore are never divisible by 3, 11 and also 13 for the rules of divisibility.

In addition, there are other factors that are NEVER divisible by all the numbers of the group - p3 - 2.


33/74

Versione 1.0

14/12/2012

Pagina 33 di 74

33

With the group p1999 - 2 there are prime numbers with:

p = 6,


p 1999^p 2 Factorizationof1999^p 2

2 3995999 7^2815513 7988005997 1301 61398974 15968023991999 31 5150975481295 3,19201E+16 756418083693357316 6,38082E+19 ,638082398400599879997 1,27553E+23 1138373324631102249116400338 2,54978E+26 736425398601302793158854857 9 5,09701E+29 4799771425328891522894723612447

10 1,01889E+33 17147591241081510171164136625616126311 2,03676E+36 74991619125964602605928592147791689544912

4,07149E+39

23

127

1393869041933111497127923285943260519

13 8,13891E+42 41532163987895190562742418641960720244186499 14 1,62697E+46 73174975502265216588683543300128811120222875447


34/74

Versione 1.0

14/12/2012

Pagina 34 di 74

34

TAB 6b:

factors frequency

7 3k+231 10k+4

Group numbers

p

1999 - 2 all end with the digit 9, 7 (the series has this cyclical).All group numbers p1999 - 2 have sum of their digits 8.

Therefore are never divisible by 3, 5, 7, 13, 19 and 29 for the rules of divisibility.

In addition, there are other factors that are NEVER divisible by all the numbers of the group - p1999 -

2.

What is important, however, is that there exist a periodicity of factors within each group so it is easier

the decomposition of very large numbers.


35/74

Versione 1.0

14/12/2012

Pagina 35 di 74

35

Examples with other groups:

p12 - 1


p 12^p -1 Factorization of12^p -1

2 143 11 133 1727 11 157

4 20735 5 11 13 29

5 248831 11 22621

6 2985983 7 11 13 19 157

7 35831807 11 659 4943

8 429981695 5 11 13 29 89 233

9 5159780351 11 37 157 80749

10 61917364223 11 13 19141 22621

TAB 7b:

p prime frequency

11 always

13 2k+2

157 3k+3

5 4k+4

22621 5k+5

The group p12 - 1 does not create NEVER any prime number, because all the number of this group are

always divisible for the factor 11


36/74

Versione 1.0

14/12/2012

Pagina 36 di 74

36

Let's see the TAB. 8:

p12 - 5

p 12^p 5 Factorizationof12^p 5

2 139 1393 1723 17234

20731

20731

5 248827 2488276 2985979 29859797 35831803 7 51188298 429981691 4299816919 5159780347 17 617 491923

10 61917364219 6191736421911 743008370683 74300837068312 8916100448251 181 4926022347113 106993205379067 7 103451 14774863114

1,28392E+15

,128391846454885915 1,5407E+16 44425273468076069

16 1,84884E+17 40146105800223201117 2,21861E+18 228119726058071721718 2,66233E+19 23946471374741639663119 3,1948E+20 795326435318116257746120 3,83376E+21 3113943988690880726141921 4,60051E+22 711565893587831153332979122 5,52061E+23 271304166988723474439294923 6,62474E+24 13722382328948561910157209124 7,94968E+25 10709742336793383050183335925 9,53962E+26 7178016488793619244786565533 26 1,14475E+28 41210213591328213075844671934127 1,37371E+29 17746900097740537855671186046728 1,64845E+30 5940011381219126271505582162031929 1,97814E+31 ,19781359483314150527412524285947 30 2,37376E+32 7391565212759411369495154305474481131 2,84852E+33 74317354702355646827293913302546371 32 3,41822E+34 179569607912829430337931119038252881170 3,48889E+75 ,


37/74

Versione 1.0

14/12/2012

Pagina 37 di 74

37

348888956932209561880025085230590467982671634313439101628517474632206804581993 2,3113E+100 2,3113E+100

TAB. 8b:

factors frequency

7 6k+1

17 16k+9

This group creates different prime numbers much more than groupp2 - 1

In fact, we have prime numbers with:

p = 2, 3, 4, 5 , 6, 8, 10 , 11, 14, 29, 70, 93

Group numbers p12 - 5 all end with the digit 9, 3, 1, 7 (the series has this cyclical).

All group numbers p12 - 5 are the sum of their digits 4.

Therefore are never divisible by 3, 5, 11, 13, 19 and 29 for the rules of divisibility.

In addition, there are other factors that are NEVER divisible by all the numbers of the group - p12 - 5.


38/74

Versione 1.0

14/12/2012

Pagina 38 di 74

38

TAB 9a:

p12 + 5

p 12^p + 5 Factorization of 12^p + 5

2 149 149

3 1733 1733

4 20741 7 2963

5 248837 23 31 3496 2985989 41 67 1087

7 35831813 47 762379

8 429981701 429981701

9 5159780357 241 647 33091

10 61917364229 7 43 983 209263

11 7,43008E+11 12211 60847463

12 8,9161E+12 83 227 473228621

13 1,06993E+14 503 1523 5783 24151

14 1,28392E+15 61 21047843681129

15 1,5407E+16 5281 291744396413316 1,84884E+17 7^2 23 487 521 4441 145589

17 2,21861E+18 17 3259 3677 35491 306857

18 2,66233E+19 718303 37064210063003

19 3,1948E+20 109 113 11119 20269 115090819

20 3,83376E+21 229 16741310010687664289

21 4,60051E+22 , 46005119909369701466117

22 5,52061E+23 7 107 737064671445175457401

23 6,62474E+24 53 257 317 2161 4401923 161288423

24 7,94968E+25 5743 13842390249589211933387

25 9,53962E+26 272693 464749 4699081 1601864461

26 1,14475E+28 , 11447545997288281555215581189


39/74

Versione 1.0

14/12/2012

Pagina 39 di 74

39

27 1,37371E+29 23 2333 292093891 8764529244253877

28 1,64845E+30 7 3263489 25859347 2790468614805121

29 1,97814E+31 61 199 8761 1763979017 105445033432399

30 2,37376E+32 47 63863 79084287742201409975992589

31 2,84852E+33 2,84852E+33

32 3,41822E+34 1081361 31610340290769550697101931701

33 4,10186E+35 17 265662474811 90824284270044415147471

34 4,92224E+36 7 587 1197915610355810830868121986644441

35 5,90668E+37

36 7,08802E+38 541381 1309247784804217077033261800856641

37 8,50562E+39 857 8243 482539 494687 70138307 71915381041697

38 1,02067E+41 23 16787 2868823 92147311622592251480162007383

39 1,22481E+42

40 1,46977E+43 7 1283 449987 3636848233340175350757842502204923

41 1,76373E+44 67043 2630738304614805036617255591214669891319

42 2,11647E+45

43 2,53977E+46

44 3,04772E+47 61

45 3,65726E+48

46 4,38871E+497 41 2447 15823

3949413660673918418259816481655385496187

47 5,26646E+50

48 6,31975E+51 1229 33345491542093916154946885481589246579451875332981

49 7,5837E+5217 23 271 15919 1910267

235355933701142664251639322060916316169

50 9,10044E+53 9,10044E+53


40/74

Versione 1.0

14/12/2012

Pagina 40 di 74

40

The corresponding table TAB. 9b is located in the next paragraph.

There are prime numbers with

p = 2, 3, 8, 21, 26, 31, 50


41/74

Versione 1.0

14/12/2012

Pagina 41 di 74

41

SUMMARY

Obviously, no group can give always consecutive prime numbers.

In general, therefore, we have:

pa b

p 2 with a integer also not prime;

b a; b must be chosen so that ( pa b) give an odd number. Better to choose at the

beginning a number b so that the first number of the group gives a prime

number.

It is therefore not necessary to have a and p prime, as for Mersenne numbers

To find out what p give rise to prime numbers, we need to build a table of the

frequencies of the factors such as the TAB. 1b.


42/74

Versione 1.0

14/12/2012

Pagina 42 di 74

42

3.4 PREDICTION OF THE FACTORS P WHICH GIVE RISE TO PRIME NUMBERS WITHIN

A GROUP pa b

To be able to "predict" in a group which is the next p which gives rise to a prime number is necessary

to construct a table of factorization of the numbers of the group, as TAB. 1b.

The factors in this table are periodic.

To give origin to a prime number within the group, the factors of the table that we have built, are

subject to the following rules:

1) When the factor is unique is a prime number Gp. From the Table 1a we can obtain the

corresponding number p that originate.

2) When we have a pair of factors, only one of the two may be a prime number within the group Gp.

Surely if there is already a known factor that gives a Gp the other cannot give rise to another Gp.

So or neither of them is, or one of them is a factor that gives rise to a prime number within the group

Gp.

3) When we have 3 or more factors, only one of the new factors give rise to a prime number Gp based

on the decomposition of the associated prime number and we must see the table in construction.

Note: only those of the first rule give p also not prime numbers, but this is possible because we derive

Gp.From rules 2 and 3 we obtain, instead, always a prime p.


43/74

Versione 1.0

14/12/2012

Pagina 43 di 74

43

For example for the group p12 +5 or the equivalent group p12 +60 that is simply "shifted of 1" with

respect to the other.

In fact:

p12 + 60 = 12 ( 1-p12 +5)

Consequently the factors of p12 +5 are simply "shifted of 1" with respect to the numbers p12 +60.

All the odd numbers of the group p12 +5 end all with the digit 9, 3, 1 and 7 (the series has this cyclical

trend). Consequently they arent never divisible by the factor 5. All the odd numbers of the groupp12 +5 have as the sum of their digits 5. Therefore they arent never divisible by 3, 11 and not even for

13, 19 and 29 for the rules of divisibility.

The table TAB. 9b is the following:

Note that the table has been intentionally left under construction because only in this way we can find

factors with a certain frequency and "predict" new Gp.

TAB. 9b:

factors frequency

7 6k+4

17 16k+17

23 11k+5

31 30k+5

41 40k+643 42k+10

47 23k+7

7^2 42k+16

53 104k+23

61 15k+14

67 330k+6

83 41k+12

101, 1,8834971362468120345740775576647e+315 294


44/74

Versione 1.0

14/12/2012

Pagina 44 di 74

44

103 204k+100107 53k+22

109 54k+19

113, 11119, 20269, 115090819 19

127, 115747627, 7,2334624677246455649146160305763e+112 114

139, 151, 263 5,4193210793765865882211347744511e+367 347

149 296k+2

167, 4,0816765767719200915746817911127e+322 301

173, 2,3848363646359265621260676968025e+306 289

181 270k+57

197, 1879,

182668222442005663634060607232966559762501915535502301012995961

199 264k+29

227, 473228621 12

229, 16741310010687664289 20241 120k+9

257, 317, 2161, 4401923, 161288423 23

269, 3061, 361183, 4,4974799275077547551990045354328e+85 90

271 270k+49

347 6,5786701127698449161492276653878e+315 295

349 5

487 324k+16

503, 1523, 5783, 24151 13521 40k+16

563 3,4872853759150355165260105110196e+345 324

569, 2,6367165468644414707750566063508e+337 316

571, 10384831, 691302947,

49253776499531718376952531684734356891526397149250317967


45/74

Versione 1.0

14/12/2012

Pagina 45 di 74

45

587 293k+34

617 2,0425365328028013701323274806668e+339 317

643, 5,5309215361761632723751858201297e+321 302

647 323k+9

653 1,0254984706772788443545658245073e+339 319

857, 8243, 482539, 494687, 70138307, 71915381041697 37

863, 3,1953908642004486090866668168237e+356 340

887 6,1091578353177625574063553696315e+347 325

983, 209263 10

991, 3,7972467484631512700551051216946e+345 323

1009 4,1828891403874064657222449978451e+332 311

1087 6

1229, 3334549, 1542093916154946885481589246579451875332981 48

1283, 449987, 3636848233340175350757842502204923 40

1733 3

2333, 292093891, 8764529244253877 27

2447, 15823, 3949413660673918418259816481655385496187 46

2963 4

3259, 3677, 35491, 306857 17

4441, 145589 16

5281, 2917443964133 15

5743, 13842390249589211933387 24


46/74

Versione 1.0

14/12/2012

Pagina 46 di 74

46

6113, 9511, 40849, 4598133134266162731878198827432870008884899 51

8761, 1763979017, 105445033432399 29

9829, 3,5942204095045724000233044826838e+109 106

12109, 3,3022237531905189371496987959636e+131 129

12211, 60847463 11

15919, 1910267, 235355933701142664251639322060916316169 49

16787, 2868823, 92147311622592251480162007383 38

33091 9

63863, 79084287742201409975992589 30

67043, 2630738304614805036617255591214669891319 41

142501 4,3181765961840496961375100657485e+114 111

272693, 464749, 4699081, 1601864461 25

541381, 1309247784804217077033261800856641 36

608483, 2,9607654251870647490509711547547e+53 57

718303, 37064210063003 18

762379 7

1081361, 31610340290769550697101931701 32

1949777, 52530409 7,5574047262360940881577088455687e+79 87

3263489, 25859347, 2790468614805121 28

429981701 8

265662474811, 90824284270044415147471 33

, 1197915610355810830868121986644441 57


47/74

Versione 1.0

14/12/2012

Pagina 47 di 74

47

factors frequency

7 6k+4

23 11k+5

61 15k+14

17 16k+1

47 23k+7

31 30k+541 40k+6

521 40k+16

83 41k+12

43 42k+10

7^2 42k+16

107 53k+22

109 54k+19

53 104k+23

241 120k+9103 204k+100

199 264k+29

271 270k+49

181 270k+57

149 296k+2

587 293k+34

647 323k+9

487 324k+16

67 330k+6


48/74

Versione 1.0

14/12/2012

Pagina 48 di 74

48

Here are some examples:

6 2985989 41 67 1087

The factor 1087 could give rise to a GP, while 41 and 67 have not given (Rule 3).

4. On some equations, lemmas and theorems concerning the expression of a number as asum of primes. [1]

Lemma 1

If ( ) 0>= YR then( ) ( ) ( )xfxfxf 21 += (4.1)

where

( ) ( ) )( )

>

++=1,

1 ...log32

nq

n xxxnxf

, (4.2)

( ) ( ) ( )+

=i

i

sdssZsY

ixf

2

2

22

1

, (4.3)

sY

has its principal value,

( )( )( )=

=h

k k

kk

sL

sLCsZ

1

', (4.4)

kC depends only on qp, and k ,

( )h

qC

=1 (4.5)

and

h

qCk . (4.6)

We note that from (4.1) and (4.3) we obtain the following expression:


49/74

Versione 1.0

14/12/2012

Pagina 49 di 74

49

( ) ( ) ( )xfxfxf 21 += ( )( )

>

++=1,

...log32

nq

nxxxn

( ) ( )+

+i

i

s dssZsYi

2

22

1

. (4.6b)

Now, we have

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

= =

=

+ =+===1, 1,,1 0

12

nq jqqj l

Yjlq

q

nejlqpjexnxfxfxf

( ) ( ) ( )( ) ( ) ( )

+

+

=++=j l

i

i

i

i

sss

qdssZsY

i

dsjlqsY

i

jlqpje

2

2

2

22

1

2

1

, (4.7)

where

( ) ( )( )

( ) ++

=j l

sqjlq

jlqpjesZ . (4.8)

Thence, we rewrite (4.7) also as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

= =

=

+ =+===1, 1,,1 0

12

nq jqqj l

Yjlq

q

n ejlqpjexnxfxfxf

( ) ( ) ( )( ) ( ) +

+

=++=j l

i

i

i

i

sss

q sYi

dsjlqsYi

jlqpje

2

2

2

22

1

2

1

( )

( )( )

dsjlq

jlqpje

j l

sq +

+. (4.8b)

Since ( ) 1, =jq , we have( )

( )( )

( )( ) =

=+

+

l

h

k k

kks

sL

sLj

hjlq

jlq

1

'1 ; (4.9)

and so

( )( )( )==

h

k k

kksL

sLCsZ

1

', (4.10)

where

( ) ( )=

=q

j

kqk jpjeh

C1

1 . (4.11)

Thence, we can rewrite the eq. (4.8b) also as follows:


50/74

Versione 1.0

14/12/2012

Pagina 50 di 74

50

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

= =

=

+ =+===1, 1,,1 0

12

nq jqqj l

Yjlq

q

n ejlqpjexnxfxfxf

( ) ( ) ( )( ) ( ) +

+

=++=j l

i

i

i

i

sss

q sYi

dsjlqsYi

jlqpje

2

2

2

22

1

2

1

( )( )

dssL

sLC

h

k k

kk

=1

'. (4.11b)

Since ( ) 0=jk if ( ) 1, >jq , the condition ( ) 1, =jq may be omitted or retained at our discretion. Thus

( ) ( )( )

( )( )

= =

===1,,1 1.,1

1

11

jqqj mqqm

qqh

qme

hpje

hC . (4.12)

Again, if 1>k we have

( ) ( )( )

( ) ( ) = =

==q

j

q

m

kqk

kqk mmeh

pjpje

hC

1 1

1

. (4.13)

If k is a primitive character,

( ) ( ) ( )=

=q

m

kkq qmme1

, , (4.14)

( ) qq k = , , (4.15)

h

qCk = . (4.16)

If is imprimitive, it belongs tod

qQ = , where 1>d . Then ( )mk has the period Q , and

( ) ( ) ( ) ( ) ( ) = =

=

=q

m

Q

n

d

l

qkqkq lQennemme1 1

1

0

. (4.17)

Lemma 2

If2

10 < , then

( )( )

=

++=h

k

kk PGChY

qxf

1

, (4.18)

where

( ) =k

YGk

, (4.19)


51/74

Versione 1.0

14/12/2012

Pagina 51 di 74

51

( )( )

+++< =

h

k

k

AYb

hqqAP

1

21

4

1211

1log , (4.20)

arctan= . (4.21)

We have, from (4.3) and (4.4),

( ) ( ) ( ) ( )( )( )

( ) +

=

+

=

===i

i

h

k

i

i

h

k

kk

k

ksks xfCdssL

sLsY

i

CdssZsY

ixf

2

2 1

2

2 1

,22

'

22

1

, (4.22)

say. But

( )( )( )

( ) ( )( )( )

+

+

+++=i

i

i

i

ssds

sL

sLsY

iYR

Yds

sL

sLsY

i

2

2

4

1

4

1

'

2

1'

2

1

b, (4.23)

where

( )( )( )

0

'

= sL

sLsYR s , (4.24)

( ){ }0sf denoting generally the residue of ( )sf for 0=s .Now

( )( )

( )( ) = =

+

+

+

+=

c c

sssL

sLss

QsL

sL

1 11 1

1'

2

1

2

1

22

1logloglog

'

aa, (4.25)

where Q is the divisor of q to which belongs, c is the number of primes which divide q but not

,...,, 21Q are the primes in question, and is a root of unity. Hence, if4

1= , we have

( )( )

( ) ( )( ) ( ) AtqAAtAqAcqAsL

sL A +++


52/74

Versione 1.0

14/12/2012

Pagina 52 di 74

52

( ) ( )( )

tse

t

tYAttYAsY

+


53/74

Versione 1.0

14/12/2012

Pagina 53 di 74

53

the path of integration being the circle Hex = , wheren

H1

= , so that

+=

2

111

nO

nx ~

n

1. (4.35)

Using the Farey dissection of order nN= , we have

( ) ( )( ) ( ) ( )( ) ( )( ) = =< ++ ===

N

q qpqp

qpqn

r

qn

r

r

qp qp

jnpeX

dXxfinpex

dxxfinv 1 1,,

,11

, ,2

1

2

1

(4.36)

say. Now

( ) ( )11121 ... +


54/74

Versione 1.0

14/12/2012

Pagina 54 di 74

54

we have:

( ) ( )

++ , A> , (4.42)

and

4

1

4

1

2

1

21

log+

=>

, AY > , (4.44)

( )


55/74

Versione 1.0

14/12/2012

Pagina 55 di 74

55

( ) ( ) ( ) ( )

+++

=


56/74

Versione 1.0

14/12/2012

Pagina 56 di 74

56

( )

+

+

++

=

++=

qp

qp qq

i

i

i

i

rr

rnYrdiO

r

nidiOdYeY

,

,

' 1

!12

, (4.54)

where

( )qN

qpqpqp

q2

1',min ,, =

< . (4.55)

Thence, from (4.53) and (4.54), we obtain also:

( ) +

+

=

++=

=

i

i

rnYri

i

nYrr

qp

q

qp

qp

diOdYeYdYeYh

q

il

,

,

'

,2

1

( )

++

=

q

diOr

ni

rr

!1

21

. (4.55b)

Also

( ) ( )


57/74

Versione 1.0

14/12/2012

Pagina 57 di 74

57

In order to complete the proof of Theorem A, we have merely to show that the finite series in (4.59)may be replaced by the infinite series rS . Now

( )( )

( ) ( ) ( )>

>


58/74

Versione 1.0

14/12/2012

Pagina 58 di 74

58

by

( )( )

( )

=

npnedwiw

enpne q

iw

q 21

2

1

. (4.66)

We are thus led to the singular series 2S . Now suppose that, instead of the integral (4.63), we consider

the integral

( ) ( ) ( )=

2

0

ReRe2

1deffRJ

kiii , (4.67)

where now k is a fixed positive integer. Instead of (4.65), we have now

( )

+

qp

qp

du

iun

iun

ekpe

kiu

q

,

,'11

~ ( ) ( )

=+

kpne

un

dukpe qq

2

2

1. (4.68)

We are thus led to suppose that

( )RJ ~( )

( )

( )

kpe

q

qn q

2

2

1

, (4.69)

when =

neR n ,

1

. Thence, we have also the following mathematical connections:

( ) ( ) ( )=

2

0

ReRe2

1deffRJ

kiii ( )

+

qp

qp

du

iun

iun

ekpe

kiu

q

,

,'11

~

~ ( ) ( )

=

+

kpne

un

dukpe qq

22

1~

( )

( )

( )

kpe

q

qn q

2

2

1

. (4.69b)

The series here (which we call 2'S ) is the singular series 2S with k in the place of n . On the other

hand

( ) ==

2

0

2loglog2

1RaRdeeReRRJ

kkiii , (4.70)

where


59/74

Versione 1.0

14/12/2012

Pagina 59 di 74

59

( )ka += loglog (4.71)

if both and k+ are prime, and oa = otherwise. Hence we obtain

2Ra ~ 22 '11

SR

. (4.72)

Here neR

1

= , but the result is easily extended to the case of continuous approach to the limit 1, andwe deduce


60/74

Versione 1.0

14/12/2012

Pagina 60 di 74

60

( )

=

=1

2

m

mxx . (4.78)

The approximation for ( ) ix = Re on qp, is

( ) iRe ~2

1

, 21

2

1

+

q

pi

nq

S qp , (4.79)

where

( )=

=q

h

qqp pheS1

2

, (4.80)

and qpS , is the conjugate of qpS , : and we find, as an approximation for ( )RJ ,

( )( )

( )

+

qp

iu

qqp

qp

qp iun

iun

duepeS

qq

q

, '

,

,

,114

1

. (4.81)

We replace the integral here by

=

+

n

iun

iun

du2

11 ; (4.82)

and we are led to the formula

( )RJ ~ Sn24

1, (4.83)

where S is the singular series( )( )

( ) =qp

qqppeS

qq

qS

,

,

. (4.84)

From the eqs. (4.77), (4.81), (4.82) and (4.83) we obtain the following relationship:

( ) ( ) ( )=

2

0

ReRe2

1defRJ

iii ( )( )

( ) =+

qp

iu

qqp

qp

qp iun

iun

duepeS

qq

q

, '

,

,

,114

1


61/74

Versione 1.0

14/12/2012

Pagina 61 di 74

61

( )( )

( ) qp

qqp peSqq

q

,

,4

1

=

+

n

iun

iun

du2

11 ~ Sn2

4

1. (4.84b)

This expression can be rewritten also as:

( ) ( ) ( )

=

2

0

ReRe12

1defRJ iii ( )( ) ( ) =

+

qp

iu

qqp

qp

qp iun

iun

duepeS

qq

q

, '

,

,

,1124

1

=

+

= n

iun

iun

du2

6116

1 ~ Sn2

24

1. (4.84c)

Thence, we conclude that the number ( )nP of primes of the form 12 +m and less than n is givenasymptotically by

( )nP ~ Sn

n

log. (4.85)

4.1On some Theorems and Equations concerning the primitive divisors of Mersennenumbers. [2]

In his interesting paper: On the sum

12

1

nd

d (1971), Erdos considered the divisor function

( ) =md

dm /11 (4.86)

restricted to Mersenne numbers; that is, numbers m of the form 12 n .

THEOREM 1.

There is a set of natural numbers S of logarithmic density 1 such that


62/74

Versione 1.0

14/12/2012

Pagina 62 di 74

62

( ) 0lim, = nnEnSn . (4.87)

With regard some equations concerning the proof of this Theorem, has been utilized the following:

( )( )( )( )

=

plqxp qPpxp

q

q xOp

p

p

p

|, ,

/11log

11, (4.88)

( )( )( )

=

=

=ndmdl

m dnA,

/1 (4.102)

so that

( ) ( )=nm

mnAnA . (4.103)

There is a Theorem that asserts that there is an 0x such that for all 0xx and any m ,

# ( ){ }

+=

x

xxxmdlxd

loglog2

logloglog3logexp: , (4.104)

or

# ( ){ }

+=

x

xxxmdlxd

2

3

log2

log3logexp: . (4.104b)

Thus by partial summation

( ) ( )( )

=

Documents

Pier Francesco Roggero, Michele Nardelli, Francesco Di Noto - PERFECT NUMBERS AND MERSENNE'S PRIME