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BIT Numerical Mathematics

, Volume 9, Issue 2, pp 125–132 | Cite as

Unique representation of primitive factors of 2 n −1,n ODD, in certain quadratic forms

  • Edgar Karst
Article

Abstract

Primitive factorsq of 2 n −1,n odd, are either of the form 8m−1 or 8m+1. In the first case there exists a unique representation ofq in the quadratic forma2−2b2, (a, b)=1,a andb odd,b<[√q/2], and in the latter a unique representation ofq in the quadratic formc2+2 j d2, (c, d)=1,c andd odd,j even and ≧6. Thus the uniqueness ofq=a2−2b2 orc2+2 j d2 exhibits a proof of the primality ofq.

A program that determinesa, b, c, andd for anyq not exceeding 16 decimal digits is described, and as an example the 13-digit prime 4432676798593 (a primitive factor of 249−1) is uniquely represented by 13742732+214·124612.

Keywords

Computational Mathematic Quadratic Form Unique Representation Decimal Digit Primitive Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© BIT Foundations 1969

Authors and Affiliations

  • Edgar Karst
    • 1
  1. 1.University of ArizonaTucsonUSA

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