Applied abstract algebra (draft) by David Joyner, Richard Kreminski, Joann Turisco

By David Joyner, Richard Kreminski, Joann Turisco

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Ap­ proximately how many 100 digit integers would have to be randomly picked before a prime is found ? (b) Estimate how many 1 00-digit prime numbers there are. 45 1 . 6. 20. Assume z is a real number greater than 1 . Let 00 ( (z) = L n- z = 1 + 2 - z + 3 - z + . . n=l , denote the Riemann zeta function. Here the sum runs over all integers n 2 1 . Let P(z) = IT (1 - p - z ) - 1 = (1 - 2 z ) - 1 (1 - 3 - zt 1 (1 - 5 - z r l . . p prime ' denote the Euler product . Here the product runs over all prime numbers p 2 2 .

1 1 again, we must have gcd( a, m) = 1 . D More generally, we have the following result. Proposition 1 . 9. Let a > 0, b > 0 and m and only if there is an integer x such that ax > 1 be integers. gcd(a, m ) l b if b ( mod m) . The result above tells us exactly when we can solve the "modulo m analogs" of the equation ax = b studied in elementary school. The proof (which requires the previous lemma and Proposition 1 . 2 . 16) is left as a good exercise. 1 . 4 Repeated squaring algorithm How hard do you think it would be to compute by hand 2 1 28 mod 5?

Using the Sieve of Eratosthenes, find all the primes from 1 to 50. 5. 14. How many digits does 269 72593 - 1 have ? 15. Check that n = 6 and n = 28 are perfect. 5 . 1 6 . Determine the prime decomposition of (a) 111, (b) 1111, (c) 1234. 5. 17. Show that if 2 n - 1 is a prime, for some integer n, then is also a prime. 18. In the notation of § 1 . 5. 1 , compute bk (p) for (a) p = 7, 1S kS p, (b) p = 11, 1 s k s p , (c) p = 13, 1S kS p . 5. 19. (a) Assume some encryption scheme requires a 100 digit prime.

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