## New Articles

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*Ref*]**definition of vector space needs no commutativity**by pahioJan 20[

*Ref*]**redundancy of two-sidedness in definition of group**by pahioJan 19[

*Ref*]**Nucleus**by porton14-12-18[

*Ref*]**spam**by vinayets1014-12-04[

*Ref*]**sequence of bounded variation**by pahio14-11-28[

*Ref*]**Numerical verification of the Goldbach conjecture**by Paulo Fernandesky14-09-28[

*Ref*]**example of contractive sequence**by pahio14-09-20[

*Ref*]**contractive sequence**by pahio14-08-29[

*Ref*]**Some formulas of partnership**by burgess14-08-26[

*Rec*]**Kenosymplirostic numbers**by imaginary.i14-08-14[

*Edu*]**How to find whether a given number is prime or not...**by burgess14-08-12[

*Edu*]**BODMAS Rule application**by burgess14-08-08[

*Edu*]**Tests of Divisibility- Simple tricks**by burgess14-08-07[

*Res*]**0/0 is possible and has an answer**by imaginary.i14-08-02## Latest Messages

10:26 am

Jan 28

Jan 26

Jan 26

Jan 25

Jan 25

Jan 25

Jan 24

Jan 22

Jan 22

Jan 16

Jan 8

Jan 8

Jan 3

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

It was not that hard to find this extensive resource regarding tetrahedron. I imagine there will be some follow up soon.
<a href="http://www.spilleautomatergratis.org/">www.spilleautomatergratis.org</a>

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 ).

Jan 25

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.

Jan 25

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

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

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.

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.

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.

Jan 8

Thanks Deva, the same to you! Now it's $5\cdot13\cdot31$.

Jan 8

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

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?

## Latest Messages

10:26 am

Jan 28

Jan 26

Jan 26

Jan 25

Jan 25

Jan 25

Jan 24

Jan 22

Jan 22

Jan 16

Jan 8

Jan 8

Jan 3

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

It was not that hard to find this extensive resource regarding tetrahedron. I imagine there will be some follow up soon.
<a href="http://www.spilleautomatergratis.org/">www.spilleautomatergratis.org</a>

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 ).

Jan 25

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.

Jan 25

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

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

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.

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.

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.

Jan 8

Thanks Deva, the same to you! Now it's $5\cdot13\cdot31$.

Jan 8

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

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?