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PlanetMath search engine

The search engine of PlanetMath is again functioning.

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?

Happy new year to all! My computer was down; today it has started functioning.Will resume posting messages at the earliest.

Thanks Deva, the same to you! Now it’s 51331normal-⋅513315\cdot 13\cdot 31.

11*31 = 341 is a pseudoprimePlanetmathPlanetmath to base 2 and 23. It is also pseudo to base (341 + i). This can be verified only if you have software pari.

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 pseudoprimePlanetmathPlanetmath 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.

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 pseudoprimePlanetmathPlanetmath to base the number + i.Example: 3*5*7 = 105. i.e. ((105+i)^104-1)/105 yields a Gaussian integer as quotient.

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 ).

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.

Hi Deva,

Speaking of congruencePlanetmathPlanetmathPlanetmath modulo a polynomialPlanetmathPlanetmath means that one considers divisibilityPlanetmathPlanetmath 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)?


Hi! The coefficients of f(x) belong to Z.

Let a + ib be a complex numberPlanetmathPlanetmath 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 pseudoprimePlanetmathPlanetmath 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 conjugatePlanetmathPlanetmath 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.

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

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