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

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 forma ²–2b ², (a, b)=1,a andb odd,b<q/2], and in the latter a unique representation ofq in the quadratic formc ²+2 ^j d², (c, d)=1,c andd odd,j even and 6. Thus the uniqueness ofq=a ²–2b ² orc ²+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.