Nerdy Science Blog

Johann Carl Friedrich Gauss was a mathematical genius born April 30^th 1777 in Braunschweig, Germany. Gauss made important discoveries in many disparate fields of science and mathematics, such as astronomy, number theory, statistics, analysis, differential geometry, electrostatics, optics, and other areas. 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 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 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), then 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 classmates 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 that have an elegant definition and many interesting properties. They are dubbed triangular numbers since they can be arranged in geometric patterns like so:

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:

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 may enjoy writing a computer program to search for more if you wish.

3) The 36^th 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 uses triangular numbers for some of its digits:

Isn’t that a bizarre 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 that 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 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. Perhaps someday I will see Gauss appear on one of their genius lists which will give it some semblance of credibility.

TABLES OF COMPUTATIONS

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, 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, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 2080, 2145, 2211, 2278, 2346, 2415, 2485, 2556, 2628, 2701, 2775, 2850, 2926, 3003, 3081, 3160, 3240, 3321, 3403, 3486, 3570, 3655, 3741, 3828, 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, 6786, 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, 10440, 10585, 10731, 10878, 11026, 11175, 11325, 11476, 11628, 11781, 11935, 12090, 12246, 12403, 12561, 12720, 12880, 13041, 13203, 13366, 13530, 13695, 13861, 14028, 14196, 14365, 14535, 14706, 14878, 15051, 15225, 15400, 15576, 15753, 15931, 16110, 16290, 16471, 16653, 16836, 17020, 17205, 17391, 17578, 17766, 17955, 18145, 18336, 18528, 18721, 18915, 19110, 19306, 19503, 19701, 19900, 20100, 20301, 20503, 20706, 20910, 21115, 21321, 21528, 21736, 21945, 22155, 22366, 22578, 22791, 23005, 23220, 23436, 23653, 23871, 24090 ,24310, 24531, 24753, 24976, 25200, 25425, 25651, 25878, 26106, 26335, 26565, 26796, 27028, 27261, 27495, 27730, 27966, 28203, 28441, 28680, 28920, 29161, 29403, 29646, 29890, 30135, 30381, 30628, 30876, 31125, 31375, 31626, 31878, 32131, 32385, 32640, 32896, 33153, 33411, 33670, 33930, 34191, 34453, 34716, 34980, 35245, 35511, 35778, 36046, 36315, 36585, 36856, 37128, 37401, 37675, 37950, 38226, 38503, 38781, 39060, 39340, 39621, 39903, 40186, 40470, 40755, 41041, 41328, 41616, 41905, 42195, 42486, 42778, 43071, 43365, 43660, 43956, 44253, 44551, 44850, 45150, 45451, 45753, 46056, 46360, 46665, 46971, 47278, 47586, 47895, 48205, 48516, 48828, 49141, 49455, 49770, 50086, 50403, 50721, 51040, 51360, 51681, 52003, 52326, 52650, 52975, 53301, 53628, 53956, 54285, 54615, 54946, 55278, 55611, 55945, 56280, 56616, 56953, 57291, 57630, 57970, 58311, 58653, 58996, 59340, 59685, 60031, 60378, 60726, 61075, 61425, 61776, 62128, 62481, 62835, 63190, 63546, 63903, 64261, 64620, 64980, 65341, 65703, 66066, 66430, 66795, 67161, 67528, 67896, 68265, 68635, 69006, 69378, 69751, 70125, 70500, 70876, 71253, 71631, 72010, 72390, 72771, 73153, 73536, 73920, 74305, 74691, 75078, 75466, 75855, 76245, 76636, 77028, 77421, 77815, 78210, 78606, 79003, 79401, 79800, 80200, 80601, 81003, 81406, 81810, 82215, 82621, 83028, 83436, 83845, 84255, 84666, 85078, 85491, 85905, 86320, 86736, 87153, 87571, 87990, 88410, 88831, 89253, 89676, 90100, 90525, 90951, 91378, 91806, 92235, 92665, 93096, 93528, 93961, 94395, 94830, 95266, 95703, 96141, 96580, 97020, 97461, 97903, 98346, 98790, 99235, 99681, 100128, 100576, 101025, 101475, 101926, 102378, 102831, 103285, 103740, 104196, 104653, 105111, 105570, 106030, 106491, 106953, 107416, 107880, 108345, 108811, 109278, 109746, 110215, 110685, 111156, 111628, 112101, 112575, 113050, 113526, 114003, 114481, 114960, 115440, 115921, 116403, 116886, 117370, 117855, 118341, 118828, 119316, 119805, 120295, 120786 ,121278, 121771, 122265, 122760, 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, 138601, 139128, 139656, 140185, 140715, 141246, 141778, 142311, 142845, 143380, 143916, 144453, 144991, 145530, 146070 ,146611, 147153, 147696, 148240, 148785, 149331, 149878, 150426, 150975, 151525, 152076, 152628, 153181, 153735, 154290, 154846, 155403, 155961,156520, 157080, 157641, 158203, 158766, 159330, 159895, 160461, 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, 178503, 179101, 179700, 180300, 180901, 181503, 182106, 182710, 183315, 183921, 184528, 185136, 185745, 186355, 186966, 187578, 188191, 188805, 189420, 190036, 190653, 191271, 191890, 192510, 193131, 193753, 194376, 195000, 195625, 196251, 196878, 197506, 198135, 198765, 199396, 200028, 200661, 201295, 201930, 202566, 203203, 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, 223446, 224115, 224785, 225456, 226128, 226801, 227475, 228150, 228826, 229503, 230181, 230860, 231540, 232221, 232903, 233586, 234270, 234955, 235641, 236328, 237016, 237705, 238395, 239086, 239778, 240471, 241165, 241860, 242556, 243253, 243951, 244650, 245350, 246051, 246753, 247456, 248160, 248865, 249571, 250278, 250986, 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, 273430, 274170, 274911, 275653, 276396, 277140, 277885, 278631, 279378, 280126, 280875, 281625, 282376, 283128, 283881, 284635, 285390, 286146, 286903, 287661, 288420, 289180, 289941, 290703, 291466, 292230, 292995, 293761, 294528, 295296, 296065, 296835, 297606, 298378, 299151, 299925, 300700, 301476, 302253, 303031, 303810, 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, 328455, 329266, 330078, 330891, 331705, 332520, 333336, 334153, 334971, 335790, 336610, 337431, 338253, 339076, 339900, 340725, 341551, 342378, 343206, 344035, 344865, 345696, 346528, 347361, 348195, 349030, 349866, 350703, 351541, 352380, 353220, 354061, 354903, 355746, 356590, 357435, 358281, 359128, 359976, 360825, 361675, 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, 388521, 389403, 390286, 391170, 392055, 392941, 393828, 394716, 395605, 396495, 397386, 398278, 399171, 400065, 400960, 401856, 402753, 403651, 404550, 405450, 406351, 407253, 408156, 409060, 409965, 410871, 411778, 412686, 413595, 414505, 415416, 416328, 417241, 418155, 419070, 419986, 420903, 421821, 422740, 423660, 424581, 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, 453628, 454581, 455535, 456490, 457446, 458403, 459361, 460320, 461280, 462241, 463203, 464166, 465130, 466095, 467061, 468028, 468996, 469965, 470935, 471906, 472878, 473851, 474825, 475800, 476776, 477753, 478731, 479710, 480690, 481671, 482653, 483636, 484620, 485605, 486591, 487578, 488566, 489555, 490545, 491536, 492528, 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

by Jason Earls

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