On the Factorization of Numbers of the Form X^2+c

Authors

  • Marc WOLF Independent researcher https://orcid.org/0000-0002-6518-9882
  • François WOLF Independent researcher https://orcid.org/0000-0002-3330-6087

DOI:

https://doi.org/10.14738/tmlai.104.12959

Keywords:

factorization, prime numbers, quadratic forms, arithmetic sequence, sequence with arithmetic difference, sieve, composite odd numbers, relationship between addition and multiplication, builder of numbers

Abstract

We study the factorization of the numbers N=X^2+c, where c is a fixed constant, and this independently of the value of gcd⁡(X,c). We prove the existence of a family of sequences with arithmetic difference (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+1. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau's 4th problem.

References

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. Marc WOLF, François WOLF (2020). Primality test and primes enumeration using odd numbers indexation. Transactions on Machine Learning and Artificial Intelligence, 8(2), 11–41. https://doi.org/10.14738/tmlai.82.8054

. Marc WOLF, François WOLF (2018), Representation theorem of composite odd numbers indices. SCIREA Journal of Mathematics, Vol. 3, pp. 106-117. http://article.scirea.org/pdf/11066.pdf.

. Marc WOLF, François WOLF, Corentin LE COZ (2018), Calculation of extended gcd by normalization. SCIREA Journal of Mathematics. Vol. 3, pp. 118-131. http://article.scirea.org/pdf/11067.pdf.

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Published

2022-08-29

How to Cite

WOLF, M., & WOLF, F. (2022). On the Factorization of Numbers of the Form X^2+c. Transactions on Machine Learning and Artificial Intelligence, 10(4), 59–77. https://doi.org/10.14738/tmlai.104.12959