A note on a conjecture of Jeśmanowicz

Moujie Deng; G. Cohen

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The Diophantine equation $x^{4} - y^{4} = z^{2}$ in three quadratic fields.

Szabó, Sándor (2004)

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On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

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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 $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).

The diophantine equation x² + 19 = yⁿ

J. H. E. Cohn (1992)

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Maohua Le (1991)

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In this article we introduce Proth numbers and prove two theorems on such numbers being prime [3]. We also give revised versions of Pocklington’s theorem and of the Legendre symbol. Finally, we prove Pepin’s theorem and that the fifth Fermat number is not prime.

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Wayne McDaniel (1993)

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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/.

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The equation $3^{a} + 5^{b} - 7^{c} = 1$ , to be solved in non-negative rational integers $a, b, c$ , has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation $y^{2} = 3^{a} + 2^{b} + 1$ , to be solved in non-negative rational integers $a, b$ and a rational integer $y$ , has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact...