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Large Language Model Performance and Clinical Reasoning Tasks.
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International Journal of Number Theory, 2011
We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts.
Knopfmacher, Arnold, Luca, Florian
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We prove that the Diophantine equation [Formula: see text] has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization [Formula: see text], then [Formula: see text] represents the number of ordered factorizations of n into prime parts.
Knopfmacher, Arnold, Luca, Florian
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1999
Es handelt sich um einen Vortrag, den der Autor im Mai 1991 vor Studenten der ETH Zürich hielt. Ohne zahlentheoretische Kenntnisse vorauszusetzen, werden zentrale Fragestellungen der Primzahltheorie vorgestellt. Den Primzahlsatz, sowie die Vermutung über die Anzahl der Primzahlzwillinge bzw.
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Es handelt sich um einen Vortrag, den der Autor im Mai 1991 vor Studenten der ETH Zürich hielt. Ohne zahlentheoretische Kenntnisse vorauszusetzen, werden zentrale Fragestellungen der Primzahltheorie vorgestellt. Den Primzahlsatz, sowie die Vermutung über die Anzahl der Primzahlzwillinge bzw.
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The College Mathematics Journal, 1994
Elementary exposition of the author's well-known survey [see mainly his monograph ``The book of prime numbers'' (Springer 1988; Zbl 0642.10001)] which has been published in slightly different versions in several languages, too.
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Elementary exposition of the author's well-known survey [see mainly his monograph ``The book of prime numbers'' (Springer 1988; Zbl 0642.10001)] which has been published in slightly different versions in several languages, too.
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American Journal of Mathematics, 1942
A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k.
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A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for which the sum of the (2m 1) -th powers of all divisors of n is not a multiple of k.
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Proceedings of the Indian Academy of Sciences - Section A, 1964
This paper, received in July 1964 and marked as being communicated by Sir C.V. Raman, then President of the Indian Academy of Sciences is one of four that DDK wrote under the Ducray pseudonym.
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This paper, received in July 1964 and marked as being communicated by Sir C.V. Raman, then President of the Indian Academy of Sciences is one of four that DDK wrote under the Ducray pseudonym.
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The American Mathematical Monthly, 2008
(2008). Prime Number Patterns. The American Mathematical Monthly: Vol. 115, No. 4, pp. 279-296.
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(2008). Prime Number Patterns. The American Mathematical Monthly: Vol. 115, No. 4, pp. 279-296.
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The Mathematical Gazette, 1961
Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument. Simple reasoning, based on the rules of probability, can however give a very fair ...
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Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument. Simple reasoning, based on the rules of probability, can however give a very fair ...
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