Enumeration of power sums modulo a prime

Andrew M. Odlyzko, Richard P. Stanley

Research output: Contribution to journalArticlepeer-review

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Abstract

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).

Original languageEnglish (US)
Pages (from-to)263-272
Number of pages10
JournalJournal of Number Theory
Volume10
Issue number2
DOIs
StatePublished - May 1978

Bibliographical note

Funding Information:
*Partially supportedb y Bell TelephoneL aboratoriesa nd by N.S.F Grant No. MCS 7308445-A04.

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