[p] **SEARCH MACHINE** by pahio Feb 20Hi admins, the search machine does not work. Please start it again!

[P] **A bit of history** by akdevaraj Feb 20Before giving further comments on Fermat's theorem and
related matters let me give a bit of history:
1640 Fermat's theorem
1740(circa) Euler's generalisation of FT
2004 Euler's generalisation of FT - a further generalisation (Devaraj))
2006 Minimum Universal exponent generalisation of Fermat's T. (Devaraj).
2012 Ultimate generalisation of FT -Pahio and Devaraj
My paper " Euler's generalisation......." freed FT of the requirement of
base and exponent to be coprime. Secondly we can identify small factors of
very large numbers by merely operating on the exponents.
Before concluding this message I would like to thank Pahio for enabling
ültimate generalisation of FT.

[P] **Euler's generalisation of Fermat's theorem in k(i) (contd)** by akdevaraj Feb 19Before generalising let me give another related example:
((15 + 7*I)^12-1)/21 also yields a Gaussian integer as quotient.

[P] **Euler's generalisation of Fermat's theorem in k(i)** by akdevaraj Feb 18I will just give an example to illustrate:
((21+i)^12-1)/21 is a Gaussian integer. Needless to say
we can verify this only if we have pari or similar
software.

**General Method for Summing Divergent Series** by Sinisa Feb 17I discovered general method for summing divergent series, which we can also consider as a method for
computing limits of divergent sequences and functions in divergent points, In this case, limits of
sequences of their partial sums. I applied the method to compute the value of some divergent integrals.
https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/>

[p] **Hi Edwards, I don't know if** by dh2718 Feb 14 Hi Edwards, I don't know if this is still actual, but here is a simple way to prove it.
Start writing down the (square of) the distance of two any points in the plane, as a function of their 4 coordinates. There are four constraints on these points. They both have to be on a given ellipse and they both have to be on a straight line of given inclination m. Now use Lagrange multipliers to maximize the distance as a function of the coordinates and the position of the line (given, for example, by its crossing point with the x axis). The rest is straightforward.

[P] **A puzzle** by akdevaraj Feb 3Fermat's theorem works in terms of square matrices; however
Euler's generalisation of Fermat's theorem in terms of matrices
does not seem to be true.

[P] **A request to Dr. Puzio** by akdevaraj Jan 30I use pari software and sometimes I would like to display
the calculations/programs on the space for messages; however, I
am unable to paste them. Would be glad if this and adding files are
enabled.

[P] **Fermat's theorem in terms of matrices.** by akdevaraj Jan 29Let X be a square matrix in which each element is an
odd prime. Then (a^(X-I)-I)/X yields a square matrix in
which the elements belong to Z. Here a is co-prime with each
element of X. Also I is the identity matrix.

[P] **pseudoprimes in k(i) (contd)- a small by-product** by akdevaraj Jan 28A small by-product of research in area of pseudoprimes in k(i):
Take a product of two numbers each with shape 4m+3. Let x be this
composite number. x is pseudo to base (x-1).Examples 21, 33, 57 etc.
(20^20-1)/21 yields a rational integer.

[p] **Rational integers and Gaussian integers** by akdevaraj Jan 26Let a + ib be a complex number where a and b belong to Z. Then a + ib is
a Gaussian integer. We get rational integers if we put b equal to 0. There
is atleast one basic difference between rational integers and Gaussian integers.
This is illustrated by the following example: 341 is a pseudoprime to base
2 and 23 i.e. (2^340-1)/341 yields a quotient which is a unique rational integer. 21
is a pseudoprime to base (21 + i ). Let ((21+i)^20-1)/21 = x. x is
a Gaussian integer; the point is x is also obtained when we change the base to
(1-21i), (-21-i) or (-1 + 21i). Hence we do not have a unique base
for obtaining x as quotient while applying Fermat's theorem. Incidentally
we get the conjugate of x when take base as (1+21i), (21-i),(-21+i) or (-1-21i).
In each of the above two cases involving x and its conjugate the four different bases
are represented by four points respectively on the complex plane.

[p] **Devaraj numbers and Carmichael numbers** by akdevaraj Jan 25Let N = p_1p_2...p_r be an r-factor composite number.If
(p_1-1)*(N-1)^(r-2)/(p_2-1)....(p_r-1) is an integer then N is
a Devaraj number. All Carmichael numbers are Devaraj numbers but the
converse is not true (see A 104016, A104017 and A166290 on OEIS ).

[p] **A property of polyomials** by akdevaraj Jan 25I might have mentioned the following property of polynomials before:
let f(x) be a polynomial in x.Then f(x+k*f(x)) is congruent to
0 (mod f(x))(here k belongs to Z. What is new in this message is that it is true
even if x is a matrix with elements being rational integers.
Also it is true even if x is a matrix with elements being Gaussian
integers.

[p] **A property of polyomials** by akdevaraj Jan 25Hi! The coefficients of f(x) belong to Z.