Non-Figurate Numbers

[modest]

Originally Posted by Turtle

off hand reviewing the list, the largest interval between pairs i see is 9, as in {...572,577...} or {...830,839...}. now if we.......... . . . . . . . . .

Interesting. Highlighting non-figs on a number chart arranged by 9's looks patternish:



I'll arrange them by 10's and edit it on to the post when done...

~modest

EDIT:

By 10's:



By 6's:

[Turtle]

Originally Posted by modest

Interesting. Highlighting non-figs on a number chart arranged by 9's looks patternish:

I'll arrange them by 10's and edit it on to the post when done...

~modest

EDIT:

By 10's:

By 6's:

interesting. very katabatakesque. all that remains is to carry on.

i just noticed that the non-figurate set contains all the primes >7! well, as far as i looked before feeling ready to jump to such a conclusion anyway. nonetheless, nothing leaped, nothing lopped i always say. moreover, the set contains composite odd numbers as well. that stickies our wickets a bit. fascinating.

[modest]

Originally Posted by Turtle

i just noticed that the non-figurate set contains all the primes >7!

I had noticed that. It also looked like the only non-figs in the set {1, 4, 7, 10, 13...} were prime {7, 13, 19, 31, 37, 43, 61... } But, 187 breaks that trend. It is non-prime and in the counting-by-threes set.

~modest

[Turtle]

Originally Posted by modest

I had noticed that. It also looked like the only non-figs in the set {1, 4, 7, 10, 13...} were prime {7, 13, 19, 31, 37, 43, 61... } But, 187 breaks that trend. It is non-prime and in the counting-by-threes set.

~modest

what is the set {1, 4, 7, 10, 13...} from?? 1,4,7 & 10 are all figurate, not non-figurate. ?

if we presume it's true that the set of primes > 7 are a proper subset of non-figurate numbers, then we can subtract them and try concentrating on non-prime non-figurate numbers to find a pattern. inasmuch as there is no known equation that gives all primes, looking for one to give all non-figurate is a complication on that problem. curiouser & more curiouser.

that's it; i'm out. . . . . . . . . . . . .

[modest]

Originally Posted by Turtle

what is the set {1, 4, 7, 10, 13...} from??

Counting by three. x = x + 3.

Originally Posted by Turtle

1,4,7 & 10 are all figurate, not non-figurate. ?

Yes. It looked as if the only non-figurate numbers in the counting by three set were prime. They are {7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151...}. Those are the only non-figurate numbers in the counting-by-three set and they all appear to be prime. But the number 187 breaks that trend. It is non-figurate, in the counting by three set, but not a prime.

If we're going to find a function or expression that gives non-figurates, I think we need to unscramble the linear sequence we have and arrange it in the way it would be output from the function. For example, I was able to find the function for this data:

6, 9, 12, 15, 18, 21, 24, 27, 30, 33...
10, 16, 22, 28, 34, 40, 46, 52, 58, 64...
15, 25, 35, 45, 55, 65, 75, 85, 95, 105...
21, 36, 51, 66, 81, 96, 111, 126, 141, 156...

because each row and each column do something predictable. That's the only way I really know how to do something like this—especially since it's clearly going to have more than one variable. But, just when I think I find the beginnings of a useful layout, it breaks. The closest I got was starting at two and counting by 12:

{2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182...}

Those are all non-figurate and in a predictable pattern which looks like it might be the top row in a useful arrangement. But, the sequence breaks at 506 which is in the counting-by-12 set but it's not a non-figurate.

If there is a function it may be beyond such a simple method.

~modest

[Jay-qu]

Well the prime issue is not so much of a surprise, as making a figurate number involves multiplication of two numbers (both of which are not 1) thus all figurate numbers can be factorised and all primes that cant be factorised must be nonfigurate.

I am sorry but I dont know anything more than you do about Diophantine equations.. Im not sure if they are applicable here, perhaps some set theory would be of some help. Ill do some research

[Turtle]

Originally Posted by Jay-qu

Well the prime issue is not so much of a surprise, as making a figurate number involves multiplication of two numbers (both of which are not 1) thus all figurate numbers can be factorised and all primes that cant be factorised must be nonfigurate.

I am sorry but I dont know anything more than you do about Diophantine equations.. Im not sure if they are applicable here, perhaps some set theory would be of some help. Ill do some research

mmm....well, you can make figurate numbers without multiplication. you can skip count for example; add every inter succesively to get triangular numbers, add every-other integer succesively to get square numbers, and so on. you can also get them by constructing a gnomon. thus the term 'figurate', sometimes called 'polygonal numbers. here is the gnomon for triangular numbers:



now in regard to the generalized equation breaking into two as a multiplier & multiplicand. in the way we have it written, the n/2 is the multiplier, but if n is odd then the term is a fracttion and so doesn't meet the criteria you lay out in your argument as we need integers. perhaps you have another 'multiplication' gives figurate number scheme than this.

anyway, i went through modestino's program-generated list & stripped out all the primes by hand. here's that set formatted in no particular fashion because i'm now dizzy. . {8,14,20,26,32,38,44,50,56,62,68,74,77,80,86,98,10 4,110,116,119,122,128,134,140,143,146,152,158,161, 164,170,182,187,188,194,200,203,206,209,212,218,22 1,224,230,236,242,248,254,266,272,278,284,290,296, 299,302,307,308,314,319,320,323,326,329,332,338,35 0,356,362,368,371,374,377,380,386,391,392,398,404, 407,410,413,416,422,434,437,440,446,452,458,464,47 0,473,476,482,488,493,494,497,500,517,518,524,527, 530,533,536,539,542,548,551,554,566,572,578,581,58 3,584,589,602,608,611,614,620,623,626,629,632,638, 644,649,650,656,662,667,668,674,686,689,692,698,70 4,707,710,713,716,722,728,731,734,737,740,746,749, 752,758,767,770,776,779,788,791,794,799,800,803,80 6,812,817,818,824,830,842,851,854,860,866,869,872, 878,884,890,893,896,899,901,902,908,913,914,917,92 0,923,926,938,943,944,950,956,959,962,968,974,979, 980,986,989,992,998 ...}

[Turtle]

just an afterthought or two that may be helpful.

in our generalized expression for figurate numbers, , n is an ordinal, and s is the number of sides. so the expression gives you the n^th s-sided number. so for example when we used s=3 and n=3 we got 6. 6 is the 3^rd 3-sided (triangular) number.

the reasons that we start with s & n> 2.

for n: whenever n=1, the equation returns F=1 regardless of what value s has. in prosaic terms, 1 is the first element of all sets of polygonal numbers. when n=2, F=s, as every number is trivially figurate. prosaically, the 2^nd 29 sided number is 29, the 2^nd 508 sided number is 508, and so on. we have to make this specification else no numbers would be non-figurate.

for s: if s=1, then F=1 regardless of the value of n. prosaically there is only 1, 1-sided number and it is 1. if s=2, then F=n regardless of the value of n. prosaically 2-sided numbers are the integers.

that's that; i'm a rat. . . . . . . . . .

[Jay-qu]

now in regard to the generalized equation breaking into two as a multiplier & multiplicand. in the way we have it written, the n/2 is the multiplier, but if n is odd then the term is a fracttion and so doesn't meet the criteria you lay out in your argument as we need integers. perhaps you have another 'multiplication' gives figurate number scheme than this

I hurried to find a proof of my hunch, here is the generalised case.

Looking back at the equation


If n is odd we have:

(odd/2)*((s-2)*odd-s+4)

two possibilities,

If S is odd,

(odd/2)*((odd-2)*odd-odd+4)

since minus 2 and plus 4 preserves oddness (and evenness)

(odd/2)*(odd*odd-odd)

since odd*odd = even

(odd/2)*(even-odd)

since even-odd = odd

(odd/2)*(odd)

since order of multiplication and division doesnt matter, this is equivalent to:

(odd*odd)/2
->
even/2 = even

obviously even cannot be prime.

The second possibility is having S even:

(odd/2)*((even-2)*odd-even+4)

using same arguments as before we have

(odd/2)*(even*odd-even)

since even*odd is always even

(odd/2)*(even-even)

even minus even will be always even, swapping order again gives

(even*odd)/2
->
even/2=even

again even cannot be prime.

So when n is odd you always get an even Figurate number, which cannot be prime.

If n is even, n/2 is an integer and thus the figurate number will have at least 1 set of factors - thus not prime.

QED

[Turtle]

Originally Posted by Jay-qu

I hurried to find a proof of my hunch, here is the generalised case.
PROOF
QED

nice. i see no blaring errors and accept the proof until otherwise notified. this gives us the authorization we need to simply subtract the set of primes from the non-figurate set as i suggested (well, went ahead & did) and then we can focus on what's left to try & find some patterns. can you guys re-write your code to do that subtraction of the primes from the non-figurate set on the fly? over......