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## Latest Messages

Apr 20
Hi Deva, perhaps the entry 'theorem on sums of two squares by Fermat' may explain it or help this problem, Jussi

Apr 19
What puzzles me is that the theorem works when the base is a prime in the ring of Gaussian integers and the exponent is a prime of shape 4m + 1 but does not work when the exponent is a prime of shape 4m+3.Can any one throw some light on this?

Apr 9

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Apr 9
In this study, we deal with functions from the square matrices to square matrices, which the same order. Such a function will be called a linear transformation, defined as follows: Let Mn(R) be a set of square matrices of order n, n ϵ S, and A be regular matrix in Mn(R), then the special function TA: Mn(R) → Mn(R) X T X A   X A   is called a linear transformation of Mn(R) to Mn(R) the following two properties are true for all X,Y ϵ Mn(R), and scalars α ϵ R: i. TA (X+Y) = TA(X) + TA(Y). (We say that TA preserves additivity) ii. TA(αX)= αTA(X) (We say that TA preserves scalar multiplication) In this case the matrix A is called the standard matrix of the function TA. Here, we transfer some well known properties of linear transformations to the above defined elements in the set all { TA: A regular in Mn(R)} [1]

Feb 27
I prefer the definition as follows: prime numbers and composite numbers in which each prime factor occurs only with exponent equal to one. Examples of such composites: 6, 35, 65, 93....

Feb 27
I prefer the definition as follows: prime numbers and composite numbers in which each prime factor occurs only with exponent equal to one. Examples of such composites: 6, 35, 65, 93....

Feb 20
So that, the first twin isolated square-free numbers are ${17,19}$, sorry.

Feb 17
This works even when the base is a Gaussian integer: Reading GPRC: gprc.txt ...Done. GP/PARI CALCULATOR Version 2.6.1 (alpha) i686 running mingw (ix86/GMP-5.0.1 kernel) 32-bit version compiled: Sep 20 2013, gcc version 4.6.3 (GCC) (readline v6.2 enabled, extended help enabled) Copyright (C) 2000-2013 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 4000000, primelimit = 500000 (17:50) gp > ((14+15*I)^104-1)/105 = -249662525598174865517621222098021785366399633335910441957688800663877876192221716937263714468906280908614454012799368615180549371243472 - 118511838209654103558982122027130965758920275429164915998560474682902951765213030198935065103035392002339412087987613469408163154998032*I (17:51) gp >

Feb 17
This works even when the base is a Gaussian integer: Reading GPRC: gprc.txt ...Done. GP/PARI CALCULATOR Version 2.6.1 (alpha) i686 running mingw (ix86/GMP-5.0.1 kernel) 32-bit version compiled: Sep 20 2013, gcc version 4.6.3 (GCC) (readline v6.2 enabled, extended help enabled) Copyright (C) 2000-2013 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?12 for how to get moral (and possibly technical) support. parisize = 4000000, primelimit = 500000 (17:50) gp > ((14+15*I)^104-1)/105 %1 = -249662525598174865517621222098021785366399633335910441957688800663877876192221716937263714468906280908614454012799368615180549371243472 - 118511838209654103558982122027130965758920275429164915998560474682902951765213030198935065103035392002339412087987613469408163154998032*I (17:51) gp >

Jan 27
Fermat's theorem works even if the base is a Gausssian integer subject to a) the prime under consideration is of shape 4m+1 and b) the exponent and base are co-prime. ((2+3*I)^16-1)/17 = -47977440 - 803040*I