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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. On February 13th 2013, PlanetMath.org was updated to use the new software system Planetary. Some release notes are here. Please report bugs in the Planetary Bugs Forum or on Github.

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[p] Devaraj numbers and Carmichael numbers by akdevaraj 3:27 am
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 ).

[p] A property of polyomials by akdevaraj 3:24 am
I 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 3:21 am
Hi! The coefficients of f(x) belong to Z.

[p] a property of polynomials by pahio Jan 24
Hi Deva, Speaking of congruence modulo a polynomial means that one considers divisibility in a certain polynomial ring. You don't specify that ring. What is it, i.e. I'm interested what kind of numbers (or others) are the coefficients of f(x)? Jussi

[p] pseudoprimes in k(i) (contd) by akdevaraj Jan 22
When we take a 3-factor composite such that two are of form 4m+3 and one is of form 4m+1 the said number is a pseudoprime to base the number + i.Example: 3*5*7 = 105. i.e. ((105+i)^104-1)/105 yields a Gaussian integer as quotient.

[p] pseudoprimes in k(i) (contd) by akdevaraj Jan 22
When we take a composite number, two integers both having shape 4m+3 , we get a composite integer which is pseudo to the base: the number umber + i. Example 21 = 3*7; this number is pseudo to the base (21 + i) i.e. ((21+i)^20-1)/21 is a Gaussian integer. In other words although 21 is not a pseudoprime in k(1), the ring of rational integers, it is pseudo to base (21+i) in the ring of Gaussian integers. This is true of all two prime factor composites(each prime having shape 4m+3). However,this can be verified only if one has pari or similar software in the computer.

[p] pseudoprimes in k(i) (contd) by akdevaraj Jan 16
11*31 = 341 is a pseudoprime to base 2 and 23. It is also pseudo to base (341 + i). This can be verified only if you have software pari.

[p] hny by pahio Jan 8
Thanks Deva, the same to you! Now it's $5\cdot13\cdot31$.

[p] hny by akdevaraj Jan 8
Happy new year to all! My computer was down; today it has started functioning.Will resume posting messages at the earliest.

[p] Disappeard old pictures by pahio Jan 3
In the PM new system, many pictures in the articles have disappeared (e.g. in the article "tractrix"). Today I saw that they may be stored in "Other useful stuff$>$Gallery". How could one transfer such pictures to their pertinent articles?

question involving sums and combinations by ogness Jan 3
m and n are integers prove that : sum(i=-m to n) (-1)^|i| * ((2m+i)! (2n-i)!)/((m+i)!^2 (n-i)!^2)) = 1 you can find it here: http://mymathforum.com/math-events/35463-sum.html>

[p] Latex formualtion ?? by rrogers 14-12-11
$$ {\displaystyle \intop_{\lambda_{1}}^{\lambda_{2}}}\phi(\lambda)\, d\lambda $$ ?? or $$ {\displaystyle \intop_{\lambda_{1}}^{\lambda_{2}}}\phi\, d\lambda $$ Answering that should answer your equestion. BTW: Try the program Lyx; it's an easy way into Latex and these symbols. Easy on-ramp and good road afterwards :)

[p] ? 2,2 2,2 by rrogers 14-12-11
Let P1=P2=S1=S2=2 (P1+P2)/(S1+S2)=1=((P1/S1)+(P2/S2))/2 I presume that your relationship symbol meant: not equal. Ray

[p] 1/0 by rrogers 14-12-11
I have seldom seen anything good coming out of division by zero.