Non-Figurate Numbers

[modest]

Originally Posted by Turtle

ok.
as an aside, and if i'm not mistaken, this is where Miss Dio-Phantine can get on stage. if it is the case that you searched your list to find 147, then suppose you had no list and still needed to find if 147, or any given integer, has an integer solution in n & s. like so:
147 = (n/2)*((s-2)*n-s+4) ?

I've gotten no closer

... still thinking on it.

~modest

[Donk]

Originally Posted by modest

I've gotten no closer

... still thinking on it.

~modest

I haven't used Diophantine equations for a lot of years, but IIRC it's a matter of looking for factors. Jay-qu did something like it a few days ago in proving that you could divide by 2 and always finish up with an integer.

I'd handle the problem this way:




294 factorises as 1,2,3,7,7. One of those factors must be n

checking n=1: . Clearly n=1 doesn't work
checking n=2: . Gives s=147. A trivial result: n=2 goes in steps of 1, generating every number.
checking n=3: . Gives 6s=300, s=50
checking n=7: . Gives 42s= 364, which is fractional and therefore not a solution.

answer: n=3, s=50.

I wrote a quick&dirty QBASIC program to generate the non-fig numbers, then looked more carefully at the figurate numbers that were being thrown away.

n = 3: 6  9  12  15  ... 3x+6
n = 4: 10  16  22  28  ... 6x+10
n = 5: 15  25  35  45  ... 10x+15
n = 6: 21  36  51  66  ... 15x+21
n = 7: 28  49  70  91  ... 21x+28
n = 8: 36  64  92  120  ... 28x+36
n = 9: 45  81  117  153  ... 36x+45
n = 10: 55  100  145  190  ... 45x+55
n = 11: 66  121  176  231  ... 55x+66
n = 12: 78  144  210  276  ... 66x+78
n = 13: 91  169  247  325  ... 78x+91
n = 14: 105  196  287  378  ... 91x+105
n = 15: 120  225  330  435  ... 105x+120
n = 16: 136  256  376  496  ... 120x+136
n = 17: 153  289  425  561  ... 136x+153 
n = 18: 171  324  477  630  ... 153x+171
n = 19: 190  361  532  703  ... 171x+190
n = 20: 210  400  590  780  ... 190x+210
n = 21: 231  441  651  861  ... 210x+231
n = 22: 253  484  715  946  ... 231x+253
n = 23: 276  529  782 ... 253x+276
n = 24: 300  576  852 ... 276x+300
n = 25: 325  625  925 ... 300x+325
n = 26: 351  676 ... 325x+351
n = 27: 378  729 ... 351x+378
n = 28: 406  784 ... 378x+406
n = 29: 435  841 ... 406x+435
n = 30: 465  900 ... 435x+465
n = 31: 496  961 ... 465x+496
n = 32: 528 
n = 33: 561 
n = 34: 595 
n = 35: 630 
n = 36: 666 
n = 37: 703 
n = 38: 741 
n = 39: 780 
n = 40: 820 
n = 41: 861 
n = 42: 903 
n = 43: 946 
n = 44: 990

For brevity, if there are more than four numbers generated I've only shown the first four and the rule they follow. Turtle's triangular numbers are very much in evidence. The whole thing is a sieve, similar to Aristophanes', which is why the primes are showing. It doesn't pick out all the composites - instead of every 7th number after 7, it removes every 21st after 28, leaving 14 and 21 in place. 21 is taken out by n=6, but 14 remains. Incidentally, if you look at s=4 you'll see that every square is taken out.

Interesting stuff

[Turtle]

Originally Posted by Donk

I haven't used Diophantine equations for a lot of years, but IIRC it's a matter of looking for factors. Jay-qu did something like it a few days ago in proving that you could divide by 2 and always finish up with an integer.

I'd handle the problem this way:




294 factorises as 1,2,3,7,7. One of those factors must be n

checking n=1: . Clearly n=1 doesn't work
checking n=2: . Gives s=147. A trivial result: n=2 goes in steps of 1, generating every number.
checking n=3: . Gives 6s=300, s=50
checking n=7: . Gives 42s= 364, which is fractional and therefore not a solution.

answer: n=3, s=50.

I wrote a quick&dirty QBASIC program to generate the non-fig numbers, then looked more carefully at the figurate numbers that were being thrown away.

n = 3: 6  9  12  15  ... 3x+6
n = 4: 10  16  22  28  ... 6x+10
n = 5: 15  25  35  45  ... 10x+15
n = 6: 21  36  51  66  ... 15x+21
...
shortened for brevity

For brevity, if there are more than four numbers generated I've only shown the first four and the rule they follow. Turtle's triangular numbers are very much in evidence. The whole thing is a sieve, similar to Aristophanes', which is why the primes are showing. It doesn't pick out all the composites - instead of every 7th number after 7, it removes every 21st after 28, leaving 14 and 21 in place. 21 is taken out by n=6, but 14 remains. Incidentally, if you look at s=4 you'll see that every square is taken out.

Interesting stuff

nothing like a new set of eyes on a problem. i actually started with the triangular numbers as a gnomon and when i started writing down the list in a table i saw , where n is the ordinal. tupling up as you did naturally to squares, the only set of powers also a set of figurate numbers, i built the gnomon, wrote down the results, and saw . for 5-sided it's , then , so on up to s=11. i derived the genralized equation from looking at those specifics.

i hear euler liked the figurates, and wrote proofs on them etcetera, however if he, or anyone else, addressed the non-figurates as a sieved out set worthy of independent consideration, i have never seen mention of it. aye, my reading is limited in relation to all what's been penned, as all such reading needs be. thank goodness for hypography and goo ol' boys such as yourself and the other respondents.

now, let me add that some figurates belong to multiple subsets, e.g. example every other hexagonal number is Triangular. for example 15 has integer roots s=3:n=5, & s=6:n=3.

so...we're on it i guess. carry on and smoke 'em if ya got 'em.

PS here's my by-hand list.

[Turtle]

continuing, one note of interest on the list i attached in the previous post. following down columns, the n index, notice that each successive s value increases by a trianagular number. for example, when n=2, then each successive s is 1 greater than the last; when n=3 each successive s is 3 greater than the last; for n=4 each successive s is 6 greater; and so on...

while the field subject of the op is the non-figurate numbers, the ground subject is the figurate numbers and no understanding what's not 'til what is is understood. just so, figurate numbers also have the name polygonal numbers and this denomination comes from the geometric arrangements of unities called gnomons. for our non-figurate, or non-polygonal numbers, it is the case that there is no such geometric arrangements that have their sum(s). (save of course the trivial n=2 arrangement).

now we have still to probe the field for diophantine or other solutions, but i continue to find the ground geometry interesting and potentially helpful in getting at the set of non-figurate numbers. that's just lipstick of course, the pig is i like constructing these gnomons and as far as i can find, no one else has bothered. ok, that's more lipstick. i made these and i'm putting them here cuz i can. oink! . . . . . . . . . .

Album of Gnomons:
Science Forums - Turtle's Album: Gnomons of Figurate/Polygonal Numbers

Gnomon of Octagonal Numbers


Just found, but not read, expose on figurate numbers:
Gnomon: from pharaohs to fractals - Google Books

[Turtle]

jaa-q's list in post #28 looks to be the so far "best", i.e. most accurate, list of the non-figurate numbers. below, his list first, then my subset of that with the primes removed. mind you J's set is correct only insomuch as his program is correct and my subset is correct only insomuch as his set is correct and my by-hand removing of primes is correct. the game's afoot. . . . . . . .

Originally Posted by Jay-qu

Non-Figurate Set

7,8,11,13,14,17,19,20,23,26,29,31,32,37,38,41,43,44,47,50,53,56,59,61,62,67,68,71,73,74,77,79,80,83
86,89,97,98,101,103,104,107,109,110,113,116,119,122,127,128,131,134,137,139,140,143,146,149,151,
152,157,158,161,163,164,167,170,173,179,181,182,187,188,191,193,194,197,199,200,203,206,209,211,
212,218,221,223,224,227,229,230,233,236,239,241,242,248,251,254,257,263,266,269,271,272,277,278,
281,283,284,290,293,296,299,302,307,308,311,313,314,317,319,320,323,326,329,331,332,337,338,347,
349,350,353,356,359,362,367,368,371,373,374,377,379,380,383,386,389,391,392,397,398,401,404,407,
409,410,413,416,419,421,422,431,433,434,437,439,440,443,446,449,452,457,458,461,463,464,467,470,
473,476,479,482,487,488,491,493,494,497,499,500,503,509,517,518,521,523,524,527,530,533,536,539,
541,542,547,548,551,554,557,563,566,569,571,572,577,578,581,583,584,587,589,593,599,601,602,607,
608,611,613,614,617,619,620,623,626,629,631,632,638,641,643,644,647,649,650,653,656,659,661,662,
667,668,673,674,677,683,686,689,691,692,698,701,704,707,709,710,713,716,719,722,727,728,731,733,734,
737,739,740,743,746,749,751,752,757,758,761,767,769,770,773,776,779,787,788,791,794,797,799,800,803
,806,809,811,812,817,818,821,823,824,827,829,830,839,842,851,853,854,857,859,860,863,866,869,872,877
,878,881,883,884,887,890,893,896,899,901,902,907,908,911,913,914,917,919,920,923,926,929,937,938,941
,943,944,947,950,953,956,959,962,967,968,971,974,977,979,980,983,986,989,991,992,997,998,

EDIT: I reran the code and fixed the above numbers.

Non-Figurate Set Sans Primes

8,14,20,26,32,38,44,50,56,62,68,74,77,80,86,98,104,110,116,119,122,128,134,140,143,
146,152,158,161,164,170,182,187,188,194,200,203,206,209,212,218,221,224,230,236,242
,248,254,266,272,278,284,290,296,299,302,308,314,319,320,323,326,329,332,338,350,356
,362,368,371,374,377,380,386,391,392,398,404,407,410,413,416,422,434,437,440,446,452
,458,464,470,473,476,482,488,493,494,497,500,517,518,524,527,530,533,536,539,542,548
,551,554,566,572,578,581,583,584,589,602,608,611,614,620,623,626,629,632,638,644,649
,650,656,662,667,668,674,686,689,692,698,704,707,710,713,716,722,728,731,734,737,740
,746,749,752,758,767,770,776,779,788,791,794,799,800,803,806,812,817,818,824,830,842
,851,854,860,866,869,872,878,884,890,893,896,899,901,902,908,913,914,917,920
,923,926,938,943,944,950,956,959,962,968,974,979,980,986,989,992,998,...

[Turtle]

Originally Posted by Turtle

Non-Figurate Set Sans Primes

8,14,20,26,32,38,44,50,56,62,68,74,77,80,86,98,104,110,116,119,122,128,134,140,143,
146,152,158,161,164,170,182,187,188,194,200,203,206,209,212,218,221,224,230,236,242
,248,254,266,272,278,284,290,296,299,302,308,314,319,320,323,326,329,332,338,350,356
,362,368,371,374,377,380,386,391,392,398,404,407,410,413,416,422,434,437,440,446,452
,458,464,470,473,476,482,488,493,494,497,500,517,518,524,527,530,533,536,539,542,548
,551,554,566,572,578,581,583,584,589,602,608,611,614,620,623,626,629,632,638,644,649
,650,656,662,667,668,674,686,689,692,698,704,707,710,713,716,722,728,731,734,737,740
,746,749,752,758,767,770,776,779,788,791,794,799,800,803,806,812,817,818,824,830,842
,851,854,860,866,869,872,878,884,890,893,896,899,901,902,908,913,914,917,920
,923,926,938,943,944,950,956,959,962,968,974,979,980,986,989,992,998,...

sum observations on the non-figurate set sans primes: . . . . .

the sum of any 2 even-non-figurate numbers is 2 more than an even-non-figurate number.
example: 20+14=34 & 34 is 2 greater than 32

the sum of an even-non-figurate number & an odd-non-figurate number is 1 less than an even-non-figurate number.
example: 20+611=631 and 631 is 1 less than 632

the sum of any 2 odd-non-figurate numbers is 2 more than an even-non-figurate number.
example: 119+ 371=490 & 490 is 2 more than 488

[Turtle]

Originally Posted by Turtle

sum observations on the non-figurate set sans primes: . . . . .

the sum of any 2 even-non-figurate numbers is 2 more than an even-non-figurate number.
example: 20+14=34 & 34 is 2 greater than 32

50+44

[Turtle]

Originally Posted by Turtle

...the sum of an even-non-figurate number & an odd-non-figurate number is 1 less than an even-non-figurate number.
example: 20+611=631 and 631 is 1 less than 632...

20+187

[freeztar]

Originally Posted by turtle

the sum of any 2 odd-non-figurate numbers is 2 more than an even-non-figurate number.
Example: 119+ 371=490 & 490 is 2 more than 488

161+187

[Turtle]

Originally Posted by freeztar

161+187

the measure of stepping advance, SA, of a turtle is given by the algebraic formula SA=2-1 and the prosaic formula, one step back for every two steps forward. . . . .