منابع مشابه
Sums and Differences of Three k-th Powers
If k ≥ 2 is a positive integer the number of representations of a positive integer N as either x1 + x k 2 = N or x k 1 − x2 = N , with integers x1 and x2, is finite. Moreover it is easily shown to be Oε(N), for any ε > 0. It is known that if k = 2 or 3 then the number of representations is unbounded as N varies, but it is conjectured that the number of representations is bounded for k ≥ 4. Inde...
متن کاملPowers in Lucas Sequences via Galois Representations
Let un be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek [6] to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then prove a general bound on admissible prime powers in a Lucas sequence assuming the Frey-Mazur conjecture on isomorphic mod p Galois representations of elliptic curves.
متن کاملNew Bounds for Gauss Sums Derived From k-th Powers, and for Heilbronn’s Exponential Sum
where p is prime, e(x) = exp(2πix), and ep(x) = e(x/p). In each case we shall assume that p | / a unless the contrary is explicitly stated. Gauss sums arise in investigations into Waring’s problem, and other additive problems involving k-th powers. Although they are amongst the simplest complete exponential sums, the question as to their true order of magnitude is far from being resolved. We re...
متن کاملExtensions of Büchi’s problem : Questions of decidability for addition and k-th powers
We generalize a question of Büchi : Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to t...
متن کاملON THE PARITY OF k-TH POWERS MOD P A GENERALIZATION OF A PROBLEM OF LEHMER
Let p be an odd prime, k, A ∈ Z, p A, d = (p − 1, k), d1 = (p − 1, k − 1), s = (p − 1)/d, t = (p − 1)/d1, E the set of even residues in Zp = Z/(p), O the set of odd residues, and Nk = #{x ∈ E : Axk ∈ O}. We give several estimates of Nk, each of different strength depending on the size and parity of s, t and k. In particular we show that Nk ∼ p4 provided that k is even and the set of all k-th po...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1972
ISSN: 0022-314X
DOI: 10.1016/0022-314x(72)90054-6