Detection of Primes in the Set of Residues of Divisors of a Given Number
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We consider the following problem: given the set of residues modulo p of all divisors of some squarefree number n, can we find efficiently a small set of residues containing all residues coming from prime factors? We present an algorithm which solves this problem for p and n satisfying some natural conditions and show that there are plenty of such numbers. One interesting feature of the proof is that it relies on additive combinatorics. We also give some application of this result to algorithmic number theory.
KeywordsSmall Doubling Probabilistic Polynomial Time Reduction Large Additional Energy Hensel Lifting Deterministic Reduction
I would like to thank Doctor Bartosz Źrałek for introducing me to the topic of deterministic reductions of integer factorization to computing values of number-theoretic functions and encouraging me to work further on the problem considered here. I would like to thank Professor Tomasz Schoen for fruitful discussions on additive combinatorics and its connection to the problem. I would also like to thank anonymous referee for suggesting a simplification of the proof of Lemma 20 which led to a better estimate and the other anonymous referee for suggestions concerning the notation, which increased the readability of the paper.
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