Triangular Numbers and the Tri Gauss Prime

Triangular Numbers and the Tri-Gauss Prime by Jason Earls, author of How to Become a Guitar Player from Hell & Heartless Bastard In Ecstasy http://becomeguitaristfromhell.blogspot.com/ http://www.youtube.com/user/zevi35711 Johann Carl Friedrich Gauss was a mathematical genius born April 30th 1777 in Braunschweig, Germany. Gauss made important discoveries in many disparate fields of mathematics, such as astronomy, number theory, statistics, analysis, differential geometry, electrostatics, and optics, among others. At an early age Gauss displayed a prodigious talent with numbers and later came to be known as the “prince of mathematicians,” and “the greatest mathematician since antiquity,” while being hugely influential to mathematicians and scientists in all areas since he contributed to so many different scientific fields. Concerning Gauss’s precocious feats of numerical skill, at the age of three he corrected errors in his father’s accounting books; and when he was ten he derived a formula for triangular numbers in a single flash of insight after his teacher challenged the class to sum all the numbers from 1 to 100; Gauss simply thought about the problem, saw the formula appear in his brilliant mind, and wrote down the correct answer (5050) and circled it. (Actually I just looked up this anecdote and found that it is not entirely true, even though that’s how it is commonly given in popular math books. The real story is that Gauss and his class were asked to sum 100 integers with a rather large difference (say 148) between each term, which is a much more difficult problem; but based on the same idea, Gauss is thought to have found the formula for triangular numbers.) Regarding triangulars, they are simple yet fascinating integers having an elegant definition and many interesting properties. They are dubbed triangular numbers since they can be arranged in geometric patterns like so: 4 4 4 4 4 4 6 4 4 4 4 4 4 4 4 4 4 10 4 1 4 4 4 3 Notice how they form (admittedly crude) triangles above. Their formula, which Gauss discovered, is T(n) = n * (n + 1)/2, which means you can insert any positive integer into the formula and a triangular number will pop out. Let’s try 16: (16 * 17)/2 = 136; so 136 is a triangular number. Here are the first 50 terms of the sequence of triangulars: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, ... And below are three of my favorite properties of triangular numbers. 1) Add any two consecutive triangulars together to produce a square. The proof can be seen geometrically like so: 4 4 4 4 4 4 + 4 4 4 4 4 4 4 4 4 4 4 4 = 3 6 9 = 32 2) Reverse the order of digits of some triangular numbers and you may get a different number that is still triangular. Here is one example: 1461195 = (1709 * 1710)/2, and reversing it produces 5911641 = (3438 * 3439)/2. These are pretty rare, but you might enjoy writing a computer program to search for more if you wish. 3) The 36th triangular number is 666, the number of the beast. 666 is one of my absolute favorite integers. It is called the number of the beast due to verse 13:18 in Revelation. Here is the Good News Bible version of that verse: This calls for wisdom. Whoever is intelligent can figure out the meaning of the number of the beast, because the number stands for the name of someone. Its number is 666. So there you have three easy and somewhat well-known properties of triangular numbers. Now let’s get a little more adventurous. Personally I like to find larger and more exotic numbers that still retain legitimate mathematical properties. I write computer programs to search for them and eventually pull the numbers straight down from the platonic realm. Here’s a prime I found that has triangular numbers as some of its digits: Tri Gauss Prime 117711771177117711771177117711771 770000000000000000000000000000077 110000000000000000000000000000011 770000000000000010000000000000077 110000000000000121000000000000011 770000000000001232100000000000077 110000000000012343210000000000011 770000000000123454321000000000077 110000000001234565432100000000011 770000000012345676543210000000077 110000000123456787654321000000011 770000001234567898765432100000077 110000000000000000000000000000011 770000000000000000000000000000077 117711771177117711771177117711771 136101521283645556678911051201361 531711902102312532763003253513784 064354654965285615956306667037417 808208619039469901035108111281176 122512751326137814311485154015961 653171117701830189119532016208021 452211227823462415248525562628270 127752850292630033081316032403321 340334863570365537413828391640054 095418642784371446545604656475348 514950505011111111111111111111111 333333333333333333333333333333333 666666666666666666666666666666666 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000001 Isn’t that a perplexing beauty? Notice the internal palindromic triangle inside the top portion of the number, which is surrounded by a border of 1177s. (I had no particular reason for including 1s and 7s other than I thought they looked good in that combination.) Beneath the bottom row of the border begins the first 100 triangular numbers concatenated together in honor of the false Gauss story. I wish I could have found a triangular number instead of a prime that had the same basic pattern seen above, but triangulars are much harder to find than primes since they have less density in the number line. The Tri Gauss Prime above is an example of ‘concrete mathematics,’ which are unusual mathematical entities I invented. Concrete math involves certain classes of numbers (squares, harshads, primes, triangulars, etc.) but with words, figures, or symbols visible in the decimal expansions to add a striking visual component to the number. The Tri Gauss Prime is a concrete prime, meaning it’s an integer having no divisors other than itself and one, while the visual component is the palindromic triangle that can be seen in the layout of the number. That is, the digits are arranged in such a way that the triangle can be “pictured” in the decimal expansion. Pretty cool, isn’t it. The Tri Gauss Prime above initially appeared in my novel, Cocoon of Terror, which was published by Afterbirth Books. (Help me out by purchasing a copy today!) Back to Johann Carl Friedrich Gauss. Gauss was also interested in philology and the study of languages and he actually had to choose between mathematics and philology when considering a career. Of course mathematics won out and he went on to make major discoveries in the field, such as being the first to prove the ‘fundamental theorem of algebra’ (although by modern standards his proof was not fully rigorous); plus he was the first to prove the quadratic reciprocity theorem (this one was legit). While still in college Gauss also proved that any regular polygon having a number of sides equal to a Fermat prime, can be constructed using only compass and straightedge, which was a major discovery. One of his later journal entries listed the line “Eureka! num = tri + tri + tri,” which meant he was the first to prove that every positive integer can be represented as the sum of at most three triangular numbers. At the age of only 21, Gauss finished his groundbreaking mathematical magnum opus titled, ‘Disquisitiones Arithmeticae’ which contained many ingenious ideas in number theory and other areas, all of which he discovered while still only a teenager. Gauss was also the first to find the basic principles of non-Euclidean geometry, but in the end decided not to published his findings. Recall that non-Euclidean geometry caused a total shift in the way math was viewed, and I suppose Gauss did not want to upset things in the math world by publishing his initial discoveries. Another accomplishment of Gauss’s was when Giuseppe Piazzi, the Italian astronomer who discovered the dwarf planet Ceres, lost sight of it for a time, and Gauss later correctly calculated its position in orbit so Piazza could locate the planet again. As I’ve been listing Gauss’s accomplishments above, I’ve been thinking of the lists of historical geniuses I have seen with guesstimates of their IQs, (that is, lists of people with the highest IQs throughout history), and how remarkable it is that Gauss has never once made an appearance on these lists. How is this possible? Gauss solved problems that no one else on Earth could even begin to contemplate; and he proved mathematical theorems that no one else could tackle, yet I have never seen his name on a list of history’s greatest geniuses. What is going on here? How can these IQ experts never mention his name, and seem to not know who he is (or any other mathematicians for that matter, besides perhaps Leibniz)? I suspect the list makers know next to nothing about mathematics or science and are biased toward literature, philosophy, and other “reading and writing” disciplines instead of “problem solving” disciplines, therefore the accomplishments of mathematicians and scientists mean nothing to them, which makes me lose faith in the validity of IQ tests and the psychologists who put them together. Come on, somebody put Gauss on one of your genius lists to give it some semblance of credibility, please. -endNOTE: TABLES OF COMPUTATIONS FOLLOW MY BIO BELOW. (Thanks for reading. If you have any comments or know of any magazines that would like to publish this article, please contact the author: zevi_35711@yahoo.com. Also you would be helping out the author greatly if you purchased one of his books from Amazon.com or another online book store of your choice. Thanks again.) http://www.youtube.com/user/zevi35711 http://becomeguitaristfromhell.blogspot.com/ http://zombiesofthereddescent.blogspot.com/ Bio: Jason Earls is the author of Cocoon of Terror (Afterbirth Books), How to Become a Guitar Player from Hell, Heartless Bastard In Ecstasy, Zombies of the Red Descent, If(Sid_Vicious == TRUE && Alan_Turing == TRUE) {ERROR_Cyberpunk(); }, Red Zen, and 0.136101521283655... all available at Amazon.com and other online book stores. His fiction and mathematical work have been published in Red Scream, Yankee Pot Roast, M-Brane SF, MathWorld.com, three of Clifford Pickover’s books, Scientia Magna, AlienSkin, Recreational and Educational Computing, Escaping Elsewhere, Neometropolis, Thirteen, Dogmatika, Prime Curios, the Online Encyclopedia of Integer Sequences, OG’s Speculative Fiction, Nocturnal Ooze, Bust Down the Door and Eat All the Chickens and other publications. He currently resides in Oklahoma with his wife, Christine. ********** TABLE ONE The First 1000 Triangular Numbers 1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210,231,253,276,3 00,325,351,378,406,435,465,496,528,561,595,630,666,703,741,780,820,861,903, 946,990,1035,1081,1128,1176,1225,1275,1326,1378,1431,1485,1540,1596,1653, 1711,1770,1830,1891,1953,2016,2080,2145,2211,2278,2346,2415,2485,2556,262 8,2701,2775,2850,2926,3003,3081,3160,3240,3321,3403,3486,3570,3655,3741,3 828,3916,4005,4095,4186,4278,4371,4465,4560,4656,4753,4851,4950,5050,5151 ,5253,5356,5460,5565,5671,5778,5886,5995,6105,6216,6328,6441,6555,6670,67 86,6903,7021,7140,7260,7381,7503,7626,7750,7875,8001,8128,8256,8385,8515, 8646,8778,8911,9045,9180,9316,9453,9591,9730,9870,10011,10153,10296,1044 0,10585,10731,10878,11026,11175,11325,11476,11628,11781,11935,12090,1224 6,12403,12561,12720,12880,13041,13203,13366,13530,13695,13861,14028,1419 6,14365,14535,14706,14878,15051,15225,15400,15576,15753,15931,16110,1629 0,16471,16653,16836,17020,17205,17391,17578,17766,17955,18145,18336,1852 8,18721,18915,19110,19306,19503,19701,19900,20100,20301,20503,20706,2091 0,21115,21321,21528,21736,21945,22155,22366,22578,22791,23005,23220,2343 6,23653,23871,24090,24310,24531,24753,24976,25200,25425,25651,25878,2610 6,26335,26565,26796,27028,27261,27495,27730,27966,28203,28441,28680,2892 0,29161,29403,29646,29890,30135,30381,30628,30876,31125,31375,31626,3187 8,32131,32385,32640,32896,33153,33411,33670,33930,34191,34453,34716,3498 0,35245,35511,35778,36046,36315,36585,36856,37128,37401,37675,37950,3822 6,38503,38781,39060,39340,39621,39903,40186,40470,40755,41041,41328,4161 6,41905,42195,42486,42778,43071,43365,43660,43956,44253,44551,44850,4515 0,45451,45753,46056,46360,46665,46971,47278,47586,47895,48205,48516,4882 8,49141,49455,49770,50086,50403,50721,51040,51360,51681,52003,52326,5265 0,52975,53301,53628,53956,54285,54615,54946,55278,55611,55945,56280,5661 6,56953,57291,57630,57970,58311,58653,58996,59340,59685,60031,60378,6072 6,61075,61425,61776,62128,62481,62835,63190,63546,63903,64261,64620,6498 0,65341,65703,66066,66430,66795,67161,67528,67896,68265,68635,69006,6937 8,69751,70125,70500,70876,71253,71631,72010,72390,72771,73153,73536,7392 0,74305,74691,75078,75466,75855,76245,76636,77028,77421,77815,78210,7860 6,79003,79401,79800,80200,80601,81003,81406,81810,82215,82621,83028,8343 6,83845,84255,84666,85078,85491,85905,86320,86736,87153,87571,87990,8841 0,88831,89253,89676,90100,90525,90951,91378,91806,92235,92665,93096,9352 8,93961,94395,94830,95266,95703,96141,96580,97020,97461,97903,98346,9879 0,99235,99681,100128,100576,101025,101475,101926,102378,102831,103285,1 03740,104196,104653,105111,105570,106030,106491,106953,107416,107880,10 8345,108811,109278,109746,110215,110685,111156,111628,112101,112575,113 050,113526,114003,114481,114960,115440,115921,116403,116886,117370,1178 55,118341,118828,119316,119805,120295,120786,121278,121771,122265,12276 0,123256,123753,124251,124750,125250,125751,126253,126756,127260,127765 ,128271,128778,129286,129795,130305,130816,131328,131841,132355,132870, 133386,133903,134421,134940,135460,135981,136503,137026,137550,138075,1 38601,139128,139656,140185,140715,141246,141778,142311,142845,143380,14 3916,144453,144991,145530,146070,146611,147153,147696,148240,148785,149 331,149878,150426,150975,151525,152076,152628,153181,153735,154290,1548 46,155403,155961,156520,157080,157641,158203,158766,159330,159895,16046 1,161028,161596,162165,162735,163306,163878,164451,165025,165600,166176 ,166753,167331,167910,168490,169071,169653,170236,170820,171405,171991, 172578,173166,173755,174345,174936,175528,176121,176715,177310,177906,1 78503,179101,179700,180300,180901,181503,182106,182710,183315,183921,18 4528,185136,185745,186355,186966,187578,188191,188805,189420,190036,190 653,191271,191890,192510,193131,193753,194376,195000,195625,196251,1968 78,197506,198135,198765,199396,200028,200661,201295,201930,202566,20320 3,203841,204480,205120,205761,206403,207046,207690,208335,208981,209628 ,210276,210925,211575,212226,212878,213531,214185,214840,215496,216153, 216811,217470,218130,218791,219453,220116,220780,221445,222111,222778,2 23446,224115,224785,225456,226128,226801,227475,228150,228826,229503,23 0181,230860,231540,232221,232903,233586,234270,234955,235641,236328,237 016,237705,238395,239086,239778,240471,241165,241860,242556,243253,2439 51,244650,245350,246051,246753,247456,248160,248865,249571,250278,25098 6,251695,252405,253116,253828,254541,255255,255970,256686,257403,258121 ,258840,259560,260281,261003,261726,262450,263175,263901,264628,265356, 266085,266815,267546,268278,269011,269745,270480,271216,271953,272691,2 73430,274170,274911,275653,276396,277140,277885,278631,279378,280126,28 0875,281625,282376,283128,283881,284635,285390,286146,286903,287661,288 420,289180,289941,290703,291466,292230,292995,293761,294528,295296,2960 65,296835,297606,298378,299151,299925,300700,301476,302253,303031,30381 0,304590,305371,306153,306936,307720,308505,309291,310078,310866,311655 ,312445,313236,314028,314821,315615,316410,317206,318003,318801,319600, 320400,321201,322003,322806,323610,324415,325221,326028,326836,327645,3 28455,329266,330078,330891,331705,332520,333336,334153,334971,335790,33 6610,337431,338253,339076,339900,340725,341551,342378,343206,344035,344 865,345696,346528,347361,348195,349030,349866,350703,351541,352380,3532 20,354061,354903,355746,356590,357435,358281,359128,359976,360825,36167 5,362526,363378,364231,365085,365940,366796,367653,368511,369370,370230 ,371091,371953,372816,373680,374545,375411,376278,377146,378015,378885, 379756,380628,381501,382375,383250,384126,385003,385881,386760,387640,3 88521,389403,390286,391170,392055,392941,393828,394716,395605,396495,39 7386,398278,399171,400065,400960,401856,402753,403651,404550,405450,406 351,407253,408156,409060,409965,410871,411778,412686,413595,414505,4154 16,416328,417241,418155,419070,419986,420903,421821,422740,423660,42458 1,425503,426426,427350,428275,429201,430128,431056,431985,432915,433846 ,434778,435711,436645,437580,438516,439453,440391,441330,442270,443211, 444153,445096,446040,446985,447931,448878,449826,450775,451725,452676,4 53628,454581,455535,456490,457446,458403,459361,460320,461280,462241,46 3203,464166,465130,466095,467061,468028,468996,469965,470935,471906,472 878,473851,474825,475800,476776,477753,478731,479710,480690,481671,4826 53,483636,484620,485605,486591,487578,488566,489555,490545,491536,49252 8,493521,494515,495510,496506,497503,498501,499500,500500, TABLE TWO Triangular Numbers Whose Reversals are also Triangular 1, 3, 6, 10, 55, 66, 120, 153, 171, 190, 300, 351, 595, 630, 666, 820, 3003, 5995, 8778, 15051, 17578, 66066, 87571, 156520, 180300, 185745, 547581, 557040, 617716, 678030, 828828, 1269621, 1461195, 1680861, 1851850, 3544453, 5073705, 5676765, 5911641, 6056940, 6295926, 12145056, 12517506, 16678200, 35133153, 56440000, 60571521, 61477416, 65054121, 157433640, 178727871, 188267310, 304119453, 354911403, 1261250200, 1264114621, 1382301910, 1634004361, 1775275491, 1945725771, 5289009825, 5329197180, 6172882716, 10246462281, 13953435931, 16048884061, 17990863516, 18226464201, 30416261403, 35615002605, 50524006140, 50620051653, 52869552900, 57003930075, 58574547585, 61536809971, 61728505930, 63410549140, 66771917766, 87350505378, 305852327670, 1205555876700, 1222080857271, 1283234841210, 1313207023131, 1664224065406, 1727580802221, 1744262774920, 3057371156400, 3549515012200, 525017 8710525, 5712769544196, 5977618167795, 6045604224661, 6149610421800, 6244562097010, 6874200024786, 6914459672175, 11700100864000, 12514986989506, 15603901754815, TABLE THREE Primes Whose Reversals are Triangular Numbers 3 19 307 523 631 1171 1801 5563 8731 12781 16831 30097 53299 54181 56629 62011 63667 64063 66457 67411 67807 108127 118801 128413 130303 131059 160453 188677 192637 196597 300583 302851 357661 506593 512011 526087 530443 533719 545473 553681 554707 557371 581041 602713 614701 618031 620731 639937 652321 657973 820117 820927 828811 870013 875269 1019503 1168831 1213633 1228393 1334233 1347877 1386199 1509463 1516483 1547101 1614331 1623907 1664083 1678843 1684387 1791793 3003967 3041803 3049507 3051901 3512323 3513007 3529441 3549547 3563731 3596599 5059783 5087143 5230171 5244751 5323321 5352337 5380849 5387383 5435533 5437711 5511061 5579641 5621113 5654017 5743099 5746753 5842981 5985829 6071761 6165919 6196123 6274549 6306697 6312043 6509341 6517999 6548653 6600259 6609331 6667291 6673627 6713101 6744583 6848029 6849541 6889411 6917761 6926347 6955777 6973651 8205643 8206201 8250787 8700031 8705269 8705791 8752717 8773111 8785171 10028917 10069903 10073269 10100953 10272511 10629253 10697941 10704259 10748827 10989397 11092069 11301463 11796121 11799217 11860777 12090979 12174931 12285901 12634651 12704347 12835531 12962251 13111561 13197601 13299679 13968091 14330467 14502331 14526973 14630401 14672773 14780323 15254119 15379921 15485401 15666661 15838399 16163929 16259833 16429663 16655383 16851007 16998301 17449147 17638237 17644681 17670421 17706151 18040969 18095527 18144901 18213931 18244711 18363781 18394507 18397549 18501121 18617653 18738703 18992341 19026127 19352503 19360459 19384993 19439353 19489753 19794241 19931941 19960687 19990297 30002023 30049651 30064429 30071161 30088099 30467953 30491911 30512809 30521521 30537361 30553291 30705427 Thanks for reading this article. Have a great day. -end-