# PlanetMath search engine

The search engine of PlanetMath is again functioning.

### Disappeard old pictures

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?

### hny

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

### hny

Thanks Deva, the same to you! Now it’s $5\cdot 13\cdot 31$.

### pseudoprimes in k(i) (contd)

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.

### pseudoprimes in k(i) (contd)

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.

### pseudoprimes in k(i) (contd)

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.

### Devaraj numbers and Carmichael numbers

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 ).

### A property of polyomials

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.

### a property of polynomials

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

### A property of polyomials

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

### Rational integers and Gaussian integers

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.

### SEARCH MACHINE

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