## Welcome!

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. On February 13th 2013, PlanetMath.org was updated to use the new software system Planetary. Some release notes are here. Please report bugs in the Planetary Bugs Forum or on Github.

## Latest Messages

8:29 pm
The answer should be in the Picard–Lindelöf theorem.

6:12 pm
The search engine of PlanetMath is again functioning.>

4:50 pm
I appreciate very much your work at PlanetMath. I am not perfect, but I am not the worst person of the planet either as this site portraits me, since I have positive things too. See \PMlinkexternal{Florentin Smarandache}{http://fs.gallup.unm.edu/FlorentinSmarandache.htm} . I have a complaint to PlanetMath and I please you to either remove or at least to update the denigration entry "FlorentinSmarandache" from PlanetMath since most things are untrue. I list them below: - For example, I did not leave my children and wife, but I brought them to America. - I have not only a doctorate degree in mathematics, but I have postdoctoral studies and researches in mathematics too (from universities in Japan, China, etc.). - I am not an Associate Professor, but a Full Professor. - I do not know who the people cited in the entry (Gregory, D. Singh, Harlestle) are. Either some ones liked to joke or tried to frame me. If somebody uses my university dynamic IP address it does not mean it is me, since there are thousands of students, faculty and staff in my campus. - I am not responsible for what other people write about my work. Also, when I put some of my papers and books in my website as anybody else, and if others copy them to other sites that’s behind my control. - My arXiv papers are not automatically transferred to math.GM, but many of my papers are in Artificial Intelligence, in Statistics, etc. (math.GM, cs.AI, math.ST, q-bio.QM). You can check it by doing a search at http://arxiv.org/ for my name. - The author (pseudo-name Torquemada, which looks to be the name of a Spanish inquisitor in the 15-16th century), unfortunately insults the arXiv (who are Cornell University professors and researchers) too by naming the math.GM (General Mathematics) as "Garbage Machine" since many scientists (not only me) from around the world submit their papers in math.GM and there are serious papers in this field. - I did not name the notions under my name, but several people who wrote about them; maybe they exaggerated so I apologize for that. - I did not do work only in elementary number theory, as per this site, but most of my research was done in applied mathematics in engineering, algebraic structures, non-Euclidean geometry and statistics. - Maybe because of some misunderstanding from both sites between some supporters and some wikipedians brought me in this horrible situation. For example I was unfortunately named in Wikipedia “criminal,” “bank robber,” “suicide bomber,” etc. –- just too much exaggerations. But I forgave them and this misunderstand has stopped long ago and I apologize for all that happened. - The cited journals in this entry were not created by me (for examples: Smarandache Notions Journal was created by Prof. C. Dumitrescu, Prof. V. S eleacu and afterwards Prof. I. B alacenoiu; Octogon Math. Mag. was created by Prof. M. Ben cze in Brasov; Progress in Physics was created by D. Ra bounski who asked me to serve as an associate editor). - The assertion that the books appear in vanity press is inaccurate, since I published in prestigious publishing houses such as: CRC Press / Taylor & Francis Group (Boca Raton - USA, London – U. K., New York - USA, Leiden - Neitherlands), IOS Press (Amsterdam – Neitherlands, Berlin – Germany, Oxford – England, Tokyo – Japan, Washington D. C. – USA), University of Applied Sciences of Zittau/Görlitz, National Defense Industry Press – Beijing, International Society for Information Fusion, ZayuPress, Pima College Press, Moorhead State University, DacoRomȃnia Tempus Comterra, Sitech (Craiova), Dacia XXI (Cluj-Napoca), Cartier - Kishinev, University of Kishinev Press, etc. - In Amazon there are not only my science books, but also books by various authors about some of my scientific research and literary creations. In Amazon there are several of my literary and art books too. - My papers were not "constantly rejected from established journals", since I have published over 250 papers, many in tens of peer-reviewed international journals of science, such as: Multiple-Valued Logic - An International Journal (now called Multiple-Valued Logic & Soft Computing), International Journal of Social Economics, International Journal of Applied Mathematics, International Journal of Tomography & Statistics, International Journal of Applied Mathematics and Statistics, International Journal of Pure and Applied Mathematics, Nieuw Archief voor Wiskunde (Holland), Intelligencer (Gottingen, Germany), Advances and Applications in Statistics, Advances in Fuzzy Sets and Systems, etc. Not talking about my tens of presentations at peer-reviewed international conferences in many countries that were published by their International Conference Proceedings. My sponsoring by NASA, NATO, U. Berkeley, SUNY IT, etc. My national and international scientific awards (New Mexico and Arizona Book Awards, Romanian Academy Award - the highest in the country, TGA Award, etc.). - People who brought positive things about my work were labelled sockpuppets in order to discourage or intimidate them unfortunately. - The entry is biased against me, since most of my research was in applied mathematics in engineering, in algebraic structures, non-Euclidean geometry – not in funny or recreational mathematics which were just one or two small papers to amuse the students. - There is nothing positive in this entry, only denigrations. I am not only bad as this entry says, I have also many positive sides (the readers should check the below site too, since I have the right to defense: http://fs.gallup.unm.edu/FlorentinSmarandache.htm ). - For more than seven years there was no updating of this entry. I suffered for more than seven years under this untrue stigma. If I committed something wrong, politically, ideologically, etc. I apologize and I humbly feel that I was punished. I am a good American citizen and I love very much America. I am now sick and getting older and I plan to retire. So please let's reconcile. Please either delete this entry or at least update it by adding positive things and removing those things that are untrue or based on speculations. I do not want to fight with anybody. I appreciate your future fairness.

4:41 pm
I appreciate very much your work at PlanetMath. I am not perfect, but I am not the worst person of the planet either as this site portraits me, since I have positive things too. See \PMlinkexternal{Florentin Smarandache}{http://fs.gallup.unm.edu/FlorentinSmarandache.htm}. I have a complaint to PlanetMath and I please you to either remove or at least to update the denigration entry "FlorentinSmarandache" from PlanetMath since most things are untrue. I list them below: - For example, I did not leave my children and wife, but I brought them to America. - I have not only a doctorate degree in mathematics, but I have postdoctoral studies and researches in mathematics too (from universities in Japan, China, etc.). - I am not an Associate Professor, but a Full Professor. - I do not know who the people cited in the entry (Gregory, D. Singh, Harlestle) are. Either some ones liked to joke or tried to frame me. If somebody uses my university dynamic IP address it does not mean it is me, since there are thousands of students, faculty and staff in my campus. - I am not responsible for what other people write about my work. Also, when I put some of my papers and books in my website as anybody else, and if others copy them to other sites that’s behind my control. - My arXiv papers are not automatically transferred to math.GM, but many of my papers are in Artificial Intelligence, in Statistics, etc. (math.GM, cs.AI, math.ST, q-bio.QM). You can check it by doing a search at http://arxiv.org/ for my name. - The author (pseudo-name Torquemada, which looks to be the name of a Spanish inquisitor in the 15-16th century), unfortunately insults the arXiv (who are Cornell University professors and researchers) too by naming the math.GM (General Mathematics) as "Garbage Machine" since many scientists (not only me) from around the world submit their papers in math.GM and there are serious papers in this field. - I did not name the notions under my name, but several people who wrote about them; maybe they exaggerated so I apologize for that. - I did not do work only in elementary number theory, as per this site, but most of my research was done in applied mathematics in engineering, algebraic structures, non-Euclidean geometry and statistics. - Maybe because of some misunderstanding from both sites between some supporters and some wikipedians brought me in this horrible situation. For example I was unfortunately named in Wikipedia “criminal,” “bank robber,” “suicide bomber,” etc. –- just too much exaggerations. But I forgave them and this misunderstand has stopped long ago and I apologize for all that happened. - The cited journals in this entry were not created by me (for examples: Smarandache Notions Journal was created by Prof. C. Dumitrescu, Prof. V. S eleacu and afterwards Prof. I. B alacenoiu; Octogon Math. Mag. was created by Prof. M. Ben cze in Brasov; Progress in Physics was created by D. Ra bounski who asked me to serve as an associate editor). - The assertion that the books appear in vanity press is inaccurate, since I published in prestigious publishing houses such as: CRC Press / Taylor & Francis Group (Boca Raton - USA, London – U. K., New York - USA, Leiden - Neitherlands), IOS Press (Amsterdam – Neitherlands, Berlin – Germany, Oxford – England, Tokyo – Japan, Washington D. C. – USA), University of Applied Sciences of Zittau/Görlitz, National Defense Industry Press – Beijing, International Society for Information Fusion, ZayuPress, Pima College Press, Moorhead State University, DacoRomȃnia Tempus Comterra, Sitech (Craiova), Dacia XXI (Cluj-Napoca), Cartier - Kishinev, University of Kishinev Press, etc. - In Amazon there are not only my science books, but also books by various authors about some of my scientific research and literary creations. In Amazon there are several of my literary and art books too. - My papers were not "constantly rejected from established journals", since I have published over 250 papers, many in tens of peer-reviewed international journals of science, such as: Multiple-Valued Logic - An International Journal (now called Multiple-Valued Logic & Soft Computing), International Journal of Social Economics, International Journal of Applied Mathematics, International Journal of Tomography & Statistics, International Journal of Applied Mathematics and Statistics, International Journal of Pure and Applied Mathematics, Nieuw Archief voor Wiskunde (Holland), Intelligencer (Gottingen, Germany), Advances and Applications in Statistics, Advances in Fuzzy Sets and Systems, etc. Not talking about my tens of presentations at peer-reviewed international conferences in many countries that were published by their International Conference Proceedings. My sponsoring by NASA, NATO, U. Berkeley, SUNY IT, etc. My national and international scientific awards (New Mexico and Arizona Book Awards, Romanian Academy Award - the highest in the country, TGA Award, etc.). - People who brought positive things about my work were labelled sockpuppets in order to discourage or intimidate them unfortunately. - The entry is biased against me, since most of my research was in applied mathematics in engineering, in algebraic structures, non-Euclidean geometry – not in funny or recreational mathematics which were just one or two small papers to amuse the students. - There is nothing positive in this entry, only denigrations. I am not only bad as this entry says, I have also many positive sides (the readers should check the below site too, since I have the right to defense: http://fs.gallup.unm.edu/FlorentinSmarandache.htm ). - For more than seven years there was no updating of this entry. I suffered for more than seven years under this unutrue stigma. If I committed something wrong, politically, ideologically, etc. I apologize and I humbly feel that I was punished. I am a good American citizen and I love very much America. I am now sick and getting older and I plan to retire. So please let's reconcile. Please either delete this entry or at least update it by adding positive things and removing those things that are untrue or based on speculations. I do not want to fight with anybody. I appreciate your future fairness.

Oct 11
I've done a few samples using direct summing and it seems that your: 1/(2^2q) should be 1/(2^q) Why don't you try it; I could have the formula feeding the summation wrong. ---------------- binomial(10,5)/(2^10); sum(binomial(2*k,k)*(-1/2)^k*binomial(10,k),k,0,10); (9/2)!/(5!*sqrt(pi)); ---------------------- Answers 63/256 63/256 63/256

Oct 11
Using Maxima's simplify_sum gives for a limit of 2*n ---- $\sum_{p=0}^{2n}\left(\frac{-1}{2}\right)^{p}\binom{2n}{p}\binom{2p}{p}$ $\frac{\left(\frac{2*n-1}{2}\right)!}{\sqrt{(\pi)}\cdot n!}$ Which seems strange until you evaluate -- $\left(\frac{\left(2\cdot4-1\right)}{2}\right)!=\left(\frac{105}{16}\right)\cdot\sqrt{\pi}$ So the sqrt(pi)'s cancel. Remember $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$ And for odd n $\sum_{p=0}^{2n+1}\left(\frac{-1}{2}\right)^{p}\binom{2n+1}{p}\binom{2p}{p}$ 0 ---------------------- Maxima also has the Zeilberger algorithm. I will copy the answer here when I understand it. If you feel you need it. -------------------- Incidently simplify_sum doesn't give a proof certificate but Zeilberger does. ---- Ray

Oct 10
Would anybody be interested in discussing a presentation here on using Pascal/Shift matrices to directly generate various Polynomial sequences via. their generating functions expressed in matrices? The underlying idea is that most generating functions are still true when indexing variables (t) are replaced with full rank singular matrices (i.e. n-1). In particular the Pascal or Shift matrices. This leads to the generating function expressed in terms of matrices and most familiar generating functions directly stating the polynomials. Also that the series is truncated automatically and exactly. I do have some theorems rather than just words :) I have a "blog" where I have stuffed some notes and results. Ray>

Oct 4
Attention Dr. Puzio: Kindly see my request to Administration. Would be glad if you would kindly do the needful.

Sep 30
This is to remind administration about my request to either a) enable " copy and paste operation" on the templates reserved for articles and messages or b) open a page on facebook which will automtically enable copying and pasting as well as uploading snapshots of articles and messages.

Sep 28
Hi parag, There is no set theory needed for the proof. I think only the expressions of m and M are strange for you; they simply mean that m is the least and M the greatest of the given fractions! Jussi

Sep 28
I am not able to understand that proof at planetmath.org/summednumeratorandsummeddenominator as I am not aware of the set theory. Please give a simple proof for that.

Sep 28
parag, do you mean such as in http://planetmath.org/summednumeratorandsummeddenominator?

Sep 28
Many of the Gaussian integers indicated in "Fermat's theorem in k(i)" and " pseudoprimes in k(i) " are too big to be copied manually. I, therefore, suggest that the template for articles and messages be modified to enable "copy and paste" operations. Alternately planetmath.org open a page on facebook. This will enable us to upload snapshots of messages/articles.

Sep 23
hi bro, i think this tip will help you : - Test for divisibility by 19. Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary. EG: 101156-->10115+2*6=10127-->1012+2*7=1026-->102+2*6=114 and 114=6*19, so 101156 is divisible by 19. and i found an harshad number digit sum 19: 874