Dept of E&C Engg, NITK Surathkal
EC445 NUMBER THEORY & APPLICATIONS IN E&C ENGG ENGG
PROBLEMS SET 2 Submission Deadline 28th October 2015 This
assignment is for 10 marks!
If you are planning to copy assignments from your friends, you
might want to think twice before doing so, cosyou will be in the
same position as you would be if you dont submit your copied
assignment! The net result is ZeroZero!!!
Fibonacci Numbers: fn = fn1+fn2 and f1 = f2 = 1. Lucas Numbers:
Ln = Ln1+Ln2 and L1 = 2, L2 = 1
Alphabets A B C D E F G H I J K L M N O P Q R S T U V W X Y
Z
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
25
1. With respect to Fibonacci sequence, establish the following
identities. You may use Mathematical Inductionwherever
applicable.A. fn fn5 = 10fn5 + fn10 B. fn+6 = 4fn+3 + fn C. For a
natural number a, fafn +fa+1fn+1 = fa+n+1. D. Binets formula for
classical Fibonacci Sequences. E. Ratio of successiveFibonacci
numbers converge to the golden mean.
2. With respect to Lucas numbers, prove the following:
A.ni=1
Li = Ln+23 B.ni=1
L2i1 = L2n2 C.ni=1
L2i = L2n+11 D.ni=1
L2i = LnLn+12
3. Express the following rational numbers into continued
fractions: A. 75/31 B. 29/8 C. 1363/422D. 50/79 E. 13/19
4. Evaluate the following continued fractions: A. 1 +1
1 + 12+ 1
2+ 13
B. 1 +1
2 + 13+ 1
4+ 15
5. Prove that the numbers 2a - 1 and 2b - 1 are relatively prime
if and only if a and b are relatively prime. What isthe
representational efficiency of a residue number system that uses
the above two moduli?
6. Obtain the representational efficiency of
RNS(15|13|11|8|7).7. Encrypt the message SECRET MESSAGE using the
keyword KRYPTOTEKNIK
8. Caeser wants to arrange a secret meeting with Marc Antony,
either at the Tiber (river) or at the Coliseum (thearena). He sends
the cipher text EVIRE. However, Antony does not know the key, so he
tries all possibilities.Where will he meet?
9. An encryption is done letter by letter using an affine cipher
with single letter keys , . Using the English alphabetso that N =
26 we find that 6 is encrypted by 23 and 15 is encrypted by 24.
What are the keys , ?
10. The cipher text SUYMC PTTWC MLCPU FRRGUG KCPWI TJQMA O was
enciphered using an affine cipher.Decipher it!
11. Decrypt the following cryptogram which was obtained using
substitution
cipher.LTOFKFKXTTEDJNLTRCPTFJSLTLFSPTTJFKXTTELTNLCVGLNMCRFXCJGNDWTNLTRCPTOFKKERTFSCJNLMCRTKNMXCCRDJNLTXTTCMNLTRCIZKPTHCJSNLTIFATWCVNLFJSFKLTKXTENLTNVRJTSFJSNVRJDJGRXXTSCJLDKEDKNCXOLDILOFKMFKNTJTSPHFXFJHFRSNCCJTORDKNNLTICSTOCRSMCRNLDKIDELTRDKFPKDJNLTProbably
a single paragraph text!
