1999/01/01 by Richard P. Brent · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #Coding theory and cryptography #Algorithm #Computer science #Artificial intelligence #Mathematics #Database
paper · doi:10.1090/s0025-5718-99-00992-8
openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
We describe the complete factorization of the tenth Fermat number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 10"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by the elliptic curve method (ECM). <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 10"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>10</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F10</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 5 comma ellipsis comma upper F 11"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">F5, … , F11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .