By Andrew Granville

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1] Suppose that this is false so there exist integers a and b that are not divisible by p, and yet ab is divisible by p. (a) Show that we may take a, b positive. Let b be the smallest such positive integer. (b) Show that 1 ≤ b < p. (c) Show that 1 < b < p. Let B be the least positive residue of p (mod b). (d) Show that 1 ≤ B < b and p|aB. ) (e) Establish a contradiction. 2. If am = bn then a/gcd(a, b) divides n. Proof. 3(a), and Am = Bn. Therefore A|Bn with (A, B) = 1, and therefore A|n by Euclid’s Lemma.

1, 4, 3, 4, 3, 1, 4, 3, 4, 3, 1 . . and the first five terms 1, 4, 3, 4, 3 repeat infinitely often. Moreover we get the same pattern if we run though the consecutive negative integer values for x. Note that in this example f (x) is never 0 or 2 (mod 5). Thus neither of the two equations x3 − 8x + 6 = 0 and x3 − 8x + 4 = 0 can have solutions in integers. 2. Let f (x) ∈ Z[x]. Suppose that f (r) ≡ 0 (mod m) for all integers r in the range 0 ≤ r ≤ m − 1. Deduce that there does not exist an integer n for which f (n) = 0.

Deduce that there does not exist an integer n for which f (n) = 0. 3 to determine all of the squares modulo m, for m = 3, 4, 5, 6, 7, 8, 9 and 10. ) (b) Show that there are no solutions in integers x, y, z to x2 + y 2 = z 2 with x and y odd. ) (c) Show that there are no solutions in integers x, y, z to x2 + y 2 = 3z 2 with (x, y) = 1. (d) Show that there are no solutions in integers x, y, z to x2 + y 2 = 6z 2 with (x, y) = 1. 4. Tests for divisibility. There are easy tests for divisibility based on ideas from this chapter.