. We describe the complete factorization of the tenth and eleventh Fermat numbers. The tenth Fermat number is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The eleventh Fermat number is a product of five prime factors with 6, 6, 21, 22 and 564 decimal digits. We also note a new 27-decimal digit factor of the thirteenth Fermat number. This number has four known prime factors and a 2391-decimal digit composite factor. All the new factors reported here were found by the elliptic curve method (ECM). The 40-digit factor of the tenth Fermat number was found after about 140 Mflop-years of computation. We discuss aspects of the practical implementation of ECM, including the use of special-purpose hardware, and note several other large factors found recently by ECM. 1. Introduction For a nonnegative integer n, the n-th Fermat number is F n = 2 2 n + 1. It is known that F n is prime for 0 n 4, and composite for 5 n 23. Also, for n 2, the factors of F n are of th...

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Abstract: The aim of this paper is to try to establish a generic model for the problem that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. More concretely, we introduced briefly the research background–the history and current situation–from Euclid’s second theorem to Green-Tao theorem. We analyzed some equivalent necessary conditions that irreducible univariable polynomials with integral coefficients represent infinitely many primes, found new necessary conditions which perhaps imply that there are only finitely many Fermat primes, generalized Euler’s function, the prime-counting function and Schinzel-Sierpinski’s Conjecture and so on, obtained an analogy of the Chinese Remainder Theorem. By proposed obtrusively several conjectures, we gave a new way for determining the existence of some special kinds of primes. Finally, we proposed sufficient and necessary conditions that several multivariable number-theoretic functions represent simultaneously primes for infinitely many integral points. Nevertheless, this is only a beginning and it miles to go. We hope that number theorists consider further it. Keywords: Euclid’s second theorem, Chinese Remainder Theorem,