Fork me on GitHub

Math for the people, by the people.


PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the Creative Commons By/Share-Alike License in order to preserve the rights of authors, readers and other content creators in a sensible way. We use LaTeX, the lingua franca of the worldwide mathematical community.

Beginning February 23th 2015 we experienced 15 days of downtime when our server stopped working. We moved a backup to DigitalOcean, and we're back online. Some features aren't working yet; we're restoring them ASAP. Please report bugs in the Planetary Bugs Forum or on Github.

User login

Error message

Notice: Undefined index: id in planetmath_legacy_urls_init() (line 5 of /home/jcorneli/beta/sites/all/modules/planetmath_legacy_urls/planetmath_legacy_urls.module).

Latest Messages  

[P] Euler's generalisation of Fermat's theorem - a further gene by akdevaraj 5:07 am
The 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 the ring of 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 3
Four 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 2
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.

[P] pseudoprimes in Z(i) (c0ntd) by akdevaraj 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.

[P] pseudoprimes in Z(i) by akdevaraj 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.

[P] Fermat's theorem in Z(i) by akdevaraj 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.

[P] vinculo-link (ECLIPSES) by JulioAlee Apr 26

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

[p] figures by pahio 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.

re: special math characters by unlord 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.>

special math characters by pahio 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!>

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

[P] A bit of history by akdevaraj 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 ├╝ltimate generalisation of FT.

[P] Euler's generalisation of Fermat's theorem in k(i) (contd) by akdevaraj Feb 19
Before generalising let me give another related example: ((15 + 7*I)^12-1)/21 also yields a Gaussian integer as quotient.