20211119, 19:05  #1 
Mar 2016
397 Posts 
factorisation with help of 2*2 matrix
A peaceful day,
I am a little bit struggled: If M is a 2*2 matrix of the form (a, b) (b, d) Let M²=E mod f, then there should be a factorisation possible: M²= (a²b², ab+bd) (ab+bd, b²d²) = E therefore (a+d)b=0, gcd (a+d, f) or gcd (b, f) should give a factor. I calculated it for M47 M= (37822, 2730) (2730, 197) M² = E mod f, but I did not get a factor. Where is the logical error ? 
20211119, 20:47  #2  
Feb 2017
Nowhere
2·3^{3}·107 Posts 
Quote:
Unfortunately, M^2 is not congruent to matid(2) modulo 2^47  1. ? M=[37822,2730;2730,197];T=M^2 %1 = [1437956584 103791870] [103791870 7491709] Now all entries of M^2 are positive, and the largest entry is < 2^31, so M^2 cannot possibly be congruent to the 2x2 identity modulo 2^47  1. Exercise: Find the largest integer m such that M^2 is congruent to matid(2) modulo m. 

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