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PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community.

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## Latest Messages

Apr 12
OK, this is not good news, but at least there is a fix. The problem has to do with the special database encoding that is needed to make these special characters work. It was set up correctly on the old server, and for some reason I thought, or rather imagined, that it would work well on the new server. More work will have to be done to fix it. However, having done it once -- in the distant past -- I can figure out how to do it again. Thanks for letting me know.>

Apr 2
Hi admins, the math fraktur letters \mathfrak{ } and the math calligraphy letters \mathcal{ } are not visible in the entries -- one sees them only as question marks (see e.g. the entry "algebraic number theory"). In the entry "ideal multiplication laws" you see much such question marks!>

Feb 20
Hi admins, the search machine does not work. Please start it again!

Feb 20
Before 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.

Feb 19
Before generalising let me give another related example: ((15 + 7*I)^12-1)/21 also yields a Gaussian integer as quotient.

Feb 18
I 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.

Feb 17
I 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/>

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.

Feb 3
Fermat'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.

Jan 30
I 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.

Jan 29
Let 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.

Jan 28
A 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.

Jan 26
Let 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.

Jan 25
Let 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 ).