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


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




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


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.


. René SCHOOF (2008), Four primality testing algorithms. Algorithmic Number Theory, MSRI Publications, Volume 44. http://www.math.leidenuniv.nl/~psh/ANTproc/05rene.pdf.

. Manindra AGRAWAL, Neeraj KAYAL, Nitin SAXENA (2004), PRIMES is in P. Ann. of Math. (2) 160, No. 2, pp. 781-793. MR2123939 (2006a:11170). http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf.

. Paul PRITCHARD (1994), Improved Incremental Prime Number Sieves. Algorithmic Number Theory Symposium. pp. 280–288. CiteSeerX http://citeseerx.ist.psu.edu/viewdoc/download?doi=

. Arthur ATKIN AND Daniel BERNSTEIN (2003), Prime sieves using binary quadratic forms. Mathematics of Computation Volume 73, Number 246, pp. 1023-1030. https://www.ams.org/journals/mcom/2004-73-246/S0025-5718-03-01501-1/S0025-5718-03-01501-1.pdf.

. 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.




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