TY - JOUR

T1 - Enumeration of power sums modulo a prime

AU - Odlyzko, Andrew M.

AU - Stanley, Richard P.

PY - 1978/5

Y1 - 1978/5

N2 - We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S⊆{1,...,p - 1} with the property that Σ∞∈Sxm ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p-12p-1 is of the order of exp(p 1 2) or less. Finally, we obtain the curious result that if p - 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ≢ 0 (mod p).

AB - We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S⊆{1,...,p - 1} with the property that Σ∞∈Sxm ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p-12p-1 is of the order of exp(p 1 2) or less. Finally, we obtain the curious result that if p - 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ≢ 0 (mod p).

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U2 - 10.1016/0022-314X(78)90039-2

DO - 10.1016/0022-314X(78)90039-2

M3 - Article

AN - SCOPUS:49349127512

VL - 10

SP - 263

EP - 272

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -