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

[p] **Fermat's theorem works when the base is a Gaussian integer** by akdevaraj Apr 19What 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?

[p] **On Measurement Assessment and Division Matrices** by ProfHasan Apr 9http://jsaer.com/download/vol-3-iss-6-2016/JSAER2016-03-06-233-237.pdf

[p] **Division of Matrices** by ProfHasan Apr 9http://jsaer.com/download/vol-3-iss-5-2016/JSAER2016-03-05-101-104.pdf

[p] **Division of Matrices** by ProfHasan Apr 9http://jsaer.com/download/vol-3-iss-5-2016/JSAER2016-03-05-101-104.pdf

[p] **Division of Matrices** by ProfHasan Apr 9http://jsaer.com/download/vol-3-iss-5-2016/JSAER2016-03-05-101-104.pdf

[p] **Division of Matrices** by ProfHasan Apr 9http://jsaer.com/download/vol-3-iss-5-2016/JSAER2016-03-05-101-104.pdf

[p] **Division of Matrices** by ProfHasan Apr 9In 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]

[P] **isolated square free numbers** by akdevaraj Feb 27I 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....

[P] **isolated square free numbers** by akdevaraj Feb 27I 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....

[P] **I forgot the number 19.** by perucho Feb 20So that, the first twin isolated square-free numbers are $
{17,19}$, sorry.

[p] **Euler's generalisation of Fermat's theorem .......** by akdevaraj Feb 17This 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 >

[p] **Euler's generalisation of Fermat's theorem .......** by akdevaraj Feb 17This 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 >

[p] **Fermat's theorem** by akdevaraj Jan 27Fermat'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