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 language | English (US) |
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Pages (from-to) | 263-272 |
Number of pages | 10 |
Journal | Journal of Number Theory |
Volume | 10 |
Issue number | 2 |
DOIs | |
State | Published - May 1978 |
Bibliographical note
Funding Information:*Partially supportedb y Bell TelephoneL aboratoriesa nd by N.S.F Grant No. MCS 7308445-A04.