The famous Fermat’s Christmas Theorem states that all Pythagorean primes can be expressed as a sum of two squares of integers. This report specifies the conditions that a (
)-type integer expressed in a sum of two squares becomes prime.
References
[1]
Zagier, D. (1990) A One-Sentence Proof That Every Prime p ≡ 1 (mod 4) Is a Sum of Two Squares. The American Mathematical Monthly, 97, 144-144. https://doi.org/10.1080/00029890.1990.11995565
[2]
Cox, D. A. (2013) Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication. Wiley. https://doi.org/ https://doi.org/10.1002/9781118400722
[3]
Davenport, H. (2008) The Higher Arithmetic: An Introduction to the Theory of Numbers. 8th Edition, Cambridge University Press. https://doi.org/10.1017/cbo9780511818097
[4]
Hardy, G.H. and Wright, E.M. (2008) An Introduction to the Theory of Numbers. Oxford University Press. https://doi.org/10.1093/oso/9780199219858.001.0001
[5]
Aletheia-Zomlefer, S.L., Fukshansky, L. and Garcia, S.R. (2020) The Bateman-Horn Conjecture: Heuristic, History, and Applications. Expositiones Mathematicae, 38, 430-479. https://doi.org/10.1016/j.exmath.2019.04.005
[6]
Iwaniec, H. and Kowalski, E. (2004) Analytic Number Theory. American Mathematical Society. https://doi.org/10.1090/coll/053
[7]
Pintz, J. (2009) Landau’s problems on primes. Journal de théorie des nombres de Bordeaux, 21, 357-404. https://doi.org/10.5802/jtnb.676
[8]
Hirschhorn, M.D. (1985) A Simple Proof of Jacobi’s Two-Square Theorem. The American MathematicalMonthly, 92, 579-580. https://doi.org/10.1080/00029890.1985.11971686
[9]
Jacobi, C. (1829) Fundamenta Nova Theoriae Functionum Ellipticarum. Borntr？ger.
