[PDF] MATH 467 FACTORIZATION AND PRIMALITY



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MATH 467 FACTORIZATION AND PRIMALITY

MATH 467 FACTORIZATION AND PRIMALITY TESTING, FALL 2017, PROBLEMS 6 Return by Monday 9th October This week’s questions require some computational aids, such as Pari or Mathemat-ica 1 Given that nis a product of two primes pand qwith p q, prove that p= n+ 1 ˚(n) p (n+ 1 ˚(n))2 4n 2:



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MATH 467 FACTORIZATION AND PRIMALITY

TESTING, FALL 2017, PROBLEMS 6

Return by Monday 9th October

This week's questions require some computational aids, such as Pari or Mathemat- ica.

1. Given thatnis a product of two primespandqwithpq, prove that

p=n+ 1(n)p(n+ 1(n))24n2 Whenn= 19749361535894833 and(n) = 19749361232517120 use this to ndp andq.

2. A \probable prime"pis a number such thatap11 (modp) fora= 2;3;5;7.

For each of the numbersnwith 100000000000n100000000025 list the ones which are probable primes and for those which are not list the values ofafor which the test fails.

3. Find allnsuch that(n) = 12.

4. Try writing a programme or script to implement Pollard's \p1" (in Pari the

exponentiation, gcd and mod algorithms are already programmed in, although for large exponents it is necessary to use the \binary expansion / successive squaring" method) and use it to factorise 950161333249.quotesdbs_dbs5.pdfusesText_10