Thursday, January 19, 2006

factoring Sylvester's sequence

Sylvester's Sequence -- From MathWorld

? a=2;for(i=0,10,print(i," ",factor(a));a=a*a-a+1)
0 Mat([2, 1])
1 Mat([3, 1])
2 Mat([7, 1])
3 Mat([43, 1])
4 [13, 1; 139, 1]
5 Mat([3263443, 1])
6 [547, 1; 607, 1; 1033, 1; 31051, 1]
7 [29881, 1; 67003, 1; 9119521, 1; 6212157481, 1]
8 [5295435634831, 1; 31401519357481261, 1; 77366930214021991992277, 1]
9 [181, 1; 1987, 1; 112374829138729, 1; 114152531605972711, 1; 35874380272246624152764569191134894955972560447869169859142453622851, 1]

10 does not look promising: small factors 2287 and 2271427 and large cofactor taken out to B1=457000 in ecm -one -I 1 -c 0 1. The algebraicity might help: it's one plus the product of all the factors seen so far. At 199-digits, or 661 bits, it's just barely within GNFS.

A091335

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