1. Show that there are infinitely many primes of the form 4n − 1. 2. Let p > 3 be a prime. Show that, when p is divided by 6, the only possible remainders are 1 and 5. 3. For what integral values of n is the number 9n3 + 33n2 + 13n − 10 3n2 + 5n − 2 an integer? 4. Let p > 3 be a prime. Show that the numerator of the sum of 1+
1 1 1 + + ··· + 2 3 p−1
is divisible by p. 5. For each positive integer n, let pn be the nth prime of the prime numbers in their natural order. Use Bertrand’s Conjecture to prove that pn < p1 + p2 + · · · + pn−1 for all n ≥ 4. 6. (PHL MO 1999) Let p1 , p2 , · · · , pn be an increasing sequence of distinct primes. Find the value of n such that 1 1 1 1+ 1+ ··· 1 + p1 p2 pn is an integer. 7. (Eotvos 1923) Prove that if the terms of an infinite arithmetic progression of natural numbers are not all equal, they cannot all be primes. 8. If one of the numbers 2n − 1 and 2n + 1 is prime, where n > 2, prove that the other is composite. 9. If p and 8p − 1 are both primes, prove that 8p + 1 is composite. 10. Prove that the square of every prime number greater than 3 yields a remainder of 1 when divided by 12. 11. If three prime numbers, all greater than 3, form an arithmetic progression, prove that the common difference is divisible by 6. 1
12. Prove that all the terms of the sequence 10001, 100010001, 1000100010001, · · · are composite. (Hint: Factor 1 + x4 + · · · + x4k using geometric series.) 13. Let p be a prime. Prove that one cannot find nonzero integers a and b √ such that a2 = pb2 . (This result implies that p is irrational.)
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