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Part of 2018 IOM

Problems(1)

Product (r_i^4 + r_j^4) modulo p

Source: IOM 2018 #3, Fedor Petrov

9/6/2018
Let kk be a positive integer such that p=8k+5p = 8k + 5 is a prime number. The integers r1,r2,…,r2k+1r_1, r_2, \dots, r_{2k+1} are chosen so that the numbers 0,r14,r24,…,r2k+140, r_1^4, r_2^4, \dots, r_{2k+1}^4 give pairwise different remainders modulo pp. Prove that the product ∏1⩽i<j⩽2k+1(ri4+rj4)\prod_{1 \leqslant i < j \leqslant 2k+1} \big(r_i^4 + r_j^4\big) is congruent to (−1)k(k+1)/2(-1)^{k(k+1)/2} modulo pp.
(Two integers are congruent modulo pp if pp divides their difference.)
Fedor Petrov
number theoryIOM