Modular Arithmetic Recap We use the notation an...a2a1a0 for the ...

2
Perfect squares
Which numbers n can be perfect squares?
Note: There are only p equivalence classes (mod p): n = 0, 1, 2, ..., or p − 1.
As n 2 ≡ (−n) 2 ≡ (p − n) 2 ( mod p), , there are at most p/2 possible values that
a perfect square can take mod p:
n 2 ≡ 0 or 1 ( mod 3),
because 02 ≡ 0 ( mod 3),
1 2 ≡ 2 2 ≡ 1 ( mod 3).
n 2 ≡ 0 or 1 ( mod 4),
because 02 ≡ 2 2 ≡ 0 ( mod 4),
1 2 ≡ 3 2 ≡ 1 ( mod 4).
n 2 ≡ 0 or 1 or −1 ( mod 5),
because 12 ≡ 4 2 ≡ 1 ( mod 4),
2 2 ≡ 3 2 ≡ −1 ( mod 4),
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n 2 ≡ 0 or 1 or 4 or 5 or 6 or 9 ( mod 10).
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If p is prime, then p 2 ≡ 1 ( mod 12) .
Indeed, looking at the factors of 12k, 12k +2, 12k +3, 12k +4, 12k +6, 12k +8, 12k +
9, 12k + 10 we see these can’t be written as p 2 for p prime.
Then 12k + 5 ≡ 12k + 11 ≡ 2 ( mod 3) so they can’t be perfect squares, and
12k + 7 ≡ 3 ( mod 4) so it can’t be a perfect square.