By Rudenskaya O. G.
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Extra resources for 3-Quasiperiodic functions on graphs and hypergraphs
To summarize, we constructed a partition of F×p into classes of four elements, with at most two exceptions having two elements each. Note that the exceptional class P1 is always present. Therefore, if p ≡ 1 (mod. 4), there must be two classes with two elements, so that −1 is a square modulo p; and if p ≡ 3 (mod. 4), there must be just one exceptional class, namely, P1 , and −1 is not a square modulo p. (ii) ⇒ (iii) Suppose that −1 is a square modulo p. So we find x ∈ Z such that p divides x 2 + 1.
Claim 2. With m, µ as before: m p = (−1)µ . Indeed, consider ( p−1)/2 (m j) = m p−1 2 ( p−1)/2 j=1 j; j=1 by Claim 1, the numbers r1 , . . , rλ , r1 , . . , rµ form a rearragement of 1, 2, . . , p−1 , so we get 2 m ( p−1)/2 p−1 2 ( p−1)/2 j ≡ (−1)µ j j=1 ( p−1)/2 j, we get m Cancelling out (mod. p). j=1 p−1 2 ≡ (−1)µ (mod. p). 1, j=1 we have m Claim 2. p−1 2 ≡ m p (mod. p), so that, finally, We now define S( p, q) = ( p−1)/2 k=0 kq p m p = (−1)µ . This proves . Claim 3. Let µ be the number of negative minimal residues of the sequence q, 2q, .
Ii) For every integer k ∈ Z, there is a unique integer r ∈ [− p/2, p/2], such that k ≡ r (mod. p); we call r the minimal residue of k. Now we numbers 2, 4, 6, . . , p − 1. compute the minimal residues of the p−1 2 Assume p ≡ 1 (mod. 4); we get 2; 4; . . ; p−1 ; − p−7 ; ; − p−3 2 2 2 . . ; −1. Note that, in absolute values, we just get a permutation of numbers 1, 2, . . , p−1 . Also, p−1 of the minimal residues the p−1 2 2 4 are negative. Taking the product of them all, we get (−1) p−1 4 ( p−1)/2 ( p−1)/2 j≡ j=1 (2 j) (mod.