New Articles
[Ref] induction proof of fundamental theorem of arithmet... by pahioMar 29
[Ref] Birkhoff Recurrence Theorem by FilipeMar 20
[Ref] Multiple Recurrence Theorem by FilipeMar 20
[Ref] convergence in the mean by pahioJan 30
[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
[Ref] Birkhoff Recurrence Theorem by FilipeMar 20
[Ref] Multiple Recurrence Theorem by FilipeMar 20
[Ref] convergence in the mean by pahioJan 30
[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
Latest Messages
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At 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.
May 1
How 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.
Apr 30
How 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.
Apr 29
If 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.
Apr 26
LINK
https://productforums.google.com/forum/#!searchin/gec-other-sentient-side/amstral/gec-other-sentient-side/iy5pWq8FhG8/gzhJ_puocWoJ
Apr 15
a/0 is the same thing as a/1
I advise you to give up this division by zero idea.
Apr 15
Dear 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.
Apr 12
OK, 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.>
Apr 2
Hi 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!>
Feb 20
Hi admins, the search machine does not work. Please start it again!
Feb 20
Before giving further comments on Fermat's theorem and
related matters let me give a bit of history:
1640 Fermat's theorem
1740(circa) Euler's generalisation of FT
2004 Euler's generalisation of FT - a further generalisation (Devaraj))
2006 Minimum Universal exponent generalisation of Fermat's T. (Devaraj).
2012 Ultimate generalisation of FT -Pahio and Devaraj
My paper " Euler's generalisation......." freed FT of the requirement of
base and exponent to be coprime. Secondly we can identify small factors of
very large numbers by merely operating on the exponents.
Before concluding this message I would like to thank Pahio for enabling
ültimate generalisation of FT.
Feb 19
Before generalising let me give another related example:
((15 + 7*I)^12-1)/21 also yields a Gaussian integer as quotient.
Feb 18
I will just give an example to illustrate:
((21+i)^12-1)/21 is a Gaussian integer. Needless to say
we can verify this only if we have pari or similar
software.
Feb 17
I discovered general method for summing divergent series, which we can also consider as a method for
computing limits of divergent sequences and functions in divergent points, In this case, limits of
sequences of their partial sums. I applied the method to compute the value of some divergent integrals.
https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/>
Latest Messages
May 2
May 1
Apr 30
Apr 29
Apr 26
Apr 15
Apr 15
Apr 12
Apr 2
Feb 20
Feb 20
Feb 19
Feb 18
Feb 17
At 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.
May 1
How 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.
Apr 30
How 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.
Apr 29
If 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.
Apr 26
LINK
https://productforums.google.com/forum/#!searchin/gec-other-sentient-side/amstral/gec-other-sentient-side/iy5pWq8FhG8/gzhJ_puocWoJ
Apr 15
a/0 is the same thing as a/1
I advise you to give up this division by zero idea.
Apr 15
Dear 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.
Apr 12
OK, 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.>
Apr 2
Hi 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!>
Feb 20
Hi admins, the search machine does not work. Please start it again!
Feb 20
Before giving further comments on Fermat's theorem and
related matters let me give a bit of history:
1640 Fermat's theorem
1740(circa) Euler's generalisation of FT
2004 Euler's generalisation of FT - a further generalisation (Devaraj))
2006 Minimum Universal exponent generalisation of Fermat's T. (Devaraj).
2012 Ultimate generalisation of FT -Pahio and Devaraj
My paper " Euler's generalisation......." freed FT of the requirement of
base and exponent to be coprime. Secondly we can identify small factors of
very large numbers by merely operating on the exponents.
Before concluding this message I would like to thank Pahio for enabling
ültimate generalisation of FT.
Feb 19
Before generalising let me give another related example:
((15 + 7*I)^12-1)/21 also yields a Gaussian integer as quotient.
Feb 18
I will just give an example to illustrate:
((21+i)^12-1)/21 is a Gaussian integer. Needless to say
we can verify this only if we have pari or similar
software.
Feb 17
I discovered general method for summing divergent series, which we can also consider as a method for
computing limits of divergent sequences and functions in divergent points, In this case, limits of
sequences of their partial sums. I applied the method to compute the value of some divergent integrals.
https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/>
