Displaying similar documents to “A note on primes p with σ ( p m ) = z n ”

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

Similarity:

W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers 2 k · 509203 (k = 1, 2,...) is of the form n - φ(n).

Lagrange’s Four-Square Theorem

Yasushige Watase (2014)

Formalized Mathematics

Similarity:

This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23]. This theorem is item #19 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

On the parity of generalized partition functions, III

Fethi Ben Saïd, Jean-Louis Nicolas, Ahlem Zekraoui (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Improving on some results of J.-L. Nicolas [], the elements of the set 𝒜 = 𝒜 ( 1 + z + z 3 + z 4 + z 5 ) , for which the partition function p ( 𝒜 , n ) (i.e. the number of partitions of n with parts in 𝒜 ) is even for all n ≥ 6 are determined. An asymptotic estimate to the counting function of this set is also given.