[P] **pseudoprimes in Z(i) (c0ntd)** by akdevaraj 10:34 amHow to create a pseudoprime with three factors in Z(i)?
Take a three-factor composite number of the following composition:
two of shape 4m+3 and one of shape 4m+1.Let N be such a composite number.
Then ((N + i)^(N-1) - 1) is congruent to 0 (mod N).Needless to
say this can be verified only if you have pari or similar software.

[P] **A Request to Administration-Reminder I** by akdevaraj May 5Facility to add files should be added. This will enable me to
show details of computation etc.

[P] **Euler's generalisation of Fermat's theorem - a further gene** by akdevaraj May 5The last word in the subject reads "generalisation". This is the
title of paper presented at Hawaii Intl conference in 2004.
Object of this message: To show this can be extended to
Gaussian integers. ((15+7*i)^20-1)/21 will yield a Gaussian integer as quotient.
((15 + 7*i)^(20+12*k)-1)/21 ; here k belongs to N. This operation
will yield a sequence of Gaussian integers. Code in pari:
{p(k)=((15+7*I)^(20+12*k)/21}

[P] **pseudoprimes in Z(i) (c0ntd)** by akdevaraj May 3Four different lattice points on the complex plane viz 15 + 7*i,
-15-7*i , -7+15*i and7-15*i produce the same latice point when
Euler's generalisation of Fermat's theorem is applied to 21.i.e.
((15+7*i)^20-1)/21 the quotient is invariant when the bases are replaced
by the other three. The conjugate of this quotient (which is a
Gaussian integer ) is the result when the base is replaced by the
conjugates of the above four points.

[P] **A request to management** by akdevaraj May 2At present there is no provision for attaching files to messages.
Request that provision be made for this. This will enable me to
post details of computations which am unable to do at present.

[P] **pseudoprimes in Z(i) (c0ntd)** by akdevaraj May 1How to find other bases which will render 21 a pseudoprime in Z(i)?
Sum of coefficients in this case is 22.Split 22 into two parts
such that one is divisible by 3 and the other by 7. Example: 15 and 7.
Then 15 +7i is a suitable base.

[P] **pseudoprimes in Z(i)** by akdevaraj Apr 30How to construct pseudoprimes in Z(i)? Let ab be a composite number where
a and b are prime. Then ab is a pseudoprime to the base
ab + i. Example: 3*7 = 21. 21 is a pseudoprime to base 21 + i.i.e.
((21 + i )^20 -1)/21 yields a Gaussian integer as quotient. Minor variations
will be illustrated in the next message.

[P] **Fermat's theorem in Z(i)** by akdevaraj Apr 29If we were to look upon the complex plane as a lattice with
horizontal lines representing real numbers and vertical ones as imaginary
ones any lattice point represents a Gaussian integer. These are
suitable bases for Fermat's theorem subject to two conditions: a)
the primes should be of form 4m + 1 b) the Gaussian integer should not be co-prime with
a prime factor of the prime chosen.
Examples: let 2 + I be the base then it is suitable for primes
17, 29 etc.

[P] **vinculo-link (ECLIPSES)** by JulioAlee Apr 26LINK
https://productforums.google.com/forum/#!searchin/gec-other-sentient-side/amstral/gec-other-sentient-side/iy5pWq8FhG8/gzhJ_puocWoJ

[P] **Pointless** by jeremyboden Apr 15a/0 is the same thing as a/1
I advise you to give up this division by zero idea.

[p] **figures** by pahio Apr 15Dear unlord,
It's nice to hear that there exists a solution to that character
problem. Of course there are here other, bigger problems -- one
of the hardest ones is perhaps formed by the figures (graphs etc.)
in the PlanetMath articles.

**re: special math characters** by unlord Apr 12OK, 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.>

**special math characters** by pahio Apr 2Hi 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!>

[p] **SEARCH MACHINE** by pahio Feb 20Hi admins, the search machine does not work. Please start it again!