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"... Geometry gives right away equations. Problem 1.1 Given an integer n is there a right triangle with rational sides and area n? The equation of such a triangle would be ..."

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Geometry gives right away equations. Problem 1.1 Given an integer n is there a right triangle with rational sides and area n? The equation of such a triangle would be

### unknown title

, 2007

"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."

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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Journal of Number Theory 128 (2008) 2398–2412 www.elsevier.com/locate/jnt Elliptic curve primality tests for Fermat and related primes

### Abstract Uncertainty can be Better than Certainty: Some Algorithms for Primality Testing ∗

"... For many years mathematicians and computer scientists have searched for a fast and reliable primality test. This is especially relevant nowadays, because the popular RSA public-key cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised alg ..."

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For many years mathematicians and computer scientists have searched for a fast and reliable primality test. This is especially relevant nowadays, because the popular RSA public-key cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. In 2002, Agrawal, Kayal and Saxena (AKS) found a deterministic polynomial-time algorithm for primality testing. I will describe the original AKS algorithm and some improvements by Bernstein and Lenstra. As far as theory is concerned, we now know that “PRIMES is in P”, and this appears to be the end of the story. However, I will explain why it is preferable to use randomised algorithms in practice.

### DETERMINISTIC ELLIPTIC CURVE PRIMALITY PROVING FOR A SPECIAL SEQUENCE OF NUMBERS

"... Abstract. We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/ ..."

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Abstract. We give a deterministic algorithm that very quickly proves the primality or compositeness of the integers N in a certain sequence, using an elliptic curve E/Q with complex multiplication by the ring of integers of Q ( √ −7). The algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit complexity that is quasi-quadratic in log N. Notably, neither of the classical “N − 1 ” or “N + 1 ” primality tests apply to the integers in our sequence. We discuss how this algorithm may be applied, in combination with sieving techniques, to efficiently search for very large primes. This has allowed us to prove the primality of several integers with more than 100,000 decimal digits, the largest of which has more than a million bits in its binary representation. We believe that this is the largest proven prime N for which no significant partial factorization of N − 1 or N + 1 is known. 1.