Abstract
In this paper, we use the analytic methods, the properties of the sixth-order characters, and the classical Gauss sums to study the computational problems of a certain special sixth residues’ modulo p and give two exact calculating formulas for them.
Highlights
Let p be an odd prime and k be a fixed positive integer
Hu and Chen [11] proved the following result: let p be an odd prime with p ≡ 7 mod 12
To get some upper bound estimates and we cannot get their exact values. In this case, we can only deduce a sharp asymptotic formula for N6(p). at is, N6(p) ird, if p is an odd prime with p ≡ 7 mod 12 and 2 is not a cubic residue modulo p, our eorem 2 obtained an exact calculating formula for N6(p), which is obviously better than the corresponding result in [11]
Summary
Let p be an odd prime and k be a fixed positive integer. For any integer a with (a, p) 1, if the congruence equation xk ≡ a mod p has a solution x, we call a is a kth residue modulo p. Legendre first introduced the characteristic function of the quadratic residues (a/p) modulo p, which later was called Legendre’s symbol. Hu and Chen [11] proved the following result: let p be an odd prime with p ≡ 7 mod 12. If 2 is a cubic residue modulo p, we have the congruence p + 4 d ≡ 11 mod 36. Ird, if p is an odd prime with p ≡ 7 mod 12 and 2 is not a cubic residue modulo p, our eorem 2 obtained an exact calculating formula for N6(p), which is obviously better than the corresponding result in [11].
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