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 2021-01-13, 07:53 #9 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 37348 Posts While the concept does not apply to Mersenne numbers since valuation(Mn,2)==1 It does apply to Fermat numbers greater than F0. The following code will find the 1st (a, b) pair for Fermat numbers greater than F0 virtually-instantly: Code: for(n=1,9,{ fermatNumber = 2^(2^n)+1; print("\nfermatNumber = ",fermatNumber ); a = sqrtint(fermatNumber -1);print("a = ",a ); b=fermatNumber -a;;print("b = ",b ); m=lift(Mod(a*b,fermatNumber );); print("F",n," >>-->> m = ",m); }) Output: Code: fermatNumber = 5 a = 2 b = 3 F1 >>-->> m = 1 fermatNumber = 17 a = 4 b = 13 F2 >>-->> m = 1 fermatNumber = 257 a = 16 b = 241 F3 >>-->> m = 1 fermatNumber = 65537 a = 256 b = 65281 F4 >>-->> m = 1 fermatNumber = 4294967297 a = 65536 b = 4294901761 F5 >>-->> m = 1 fermatNumber = 18446744073709551617 a = 4294967296 b = 18446744069414584321 F6 >>-->> m = 1 fermatNumber = 340282366920938463463374607431768211457 a = 18446744073709551616 b = 340282366920938463444927863358058659841 F7 >>-->> m = 1 fermatNumber = 115792089237316195423570985008687907853269984665640564039457584007913129639937 a = 340282366920938463463374607431768211456 b = 115792089237316195423570985008687907852929702298719625575994209400481361428481 F8 >>-->> m = 1 fermatNumber = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097 a = 115792089237316195423570985008687907853269984665640564039457584007913129639936 b = 13407807929942597099574024998205846127479365820592393377723561443721764030073431184712636981971479856705023170278632780869088242247907112362425735876444161 F9 >>-->> m = 1 For F1 to F4 there are no other positive integer pairs (a, b) other than listed. For F5 however there is a 2nd pair: Code: \\DTC-120-A From Rashid Naimi - 1/13/2321 BC F5 = 2^(2^5)+1 a = 46837383 b = F5-a (a*b-1)/F5 Output: Code: (02:48) gp > F5 = 2^(2^5)+1 %31 = 4294967297 (02:48) gp > a = 46837383 %32 = 46837383 (02:48) gp > b = F5-a %33 = 4248129914 (02:48) gp > (a*b-1)/F5 %34 = 46326613 Unfortunately I have no clue how to find the secondary (a, b) pairs without Bruce-Lee Brute-Force. Thank you for your time.