# Is every positive integer the sum of 1 square number, 1 pentagonal number, and 1 hexagonal number?

I found this interesting conjecture, but maybe I'm not the first to state it. I have tested it for the first $10^4$ positive integers, but that is not a proof. Can anybody prove or disprove this conjecture?

Every positive integer can be written as the sum of 1 square number, 1 pentagonal number, and 1 hexagonal number.

Note:

Square numbers are generated by the formula, $S_{n}=n^{2}$. The first ten square numbers are:

0, 1, 4, 9, 16, 25, 36, 49, 64, 81,...


Pentagonal numbers are generated by the formula, $P_{n}=\frac{1}{2}n(3n-1)$. The first ten pentagonal numbers are:

0, 1, 5, 12, 22, 35, 51, 70, 92, 117,...


Hexagonal numbers are generated by the formula, $H_{n} = n(2n-1)$. The first ten Hexagonal numbers are:

0, 1, 6, 15, 28, 45, 66, 91, 120, 153,...


Here are the solutions for the first 10 positive integers.

Numbers = Square + Pentagon + Hexagon

1    =    0    +    1    +    0
2    =    1    +    1    +    0
3    =    1    +    1    +    1
4    =    4    +    0    +    0
5    =    0    +    5    +    0
6    =    1    +    5    +    0
7    =    1    +    5    +    1
8    =    1    +    1    +    6
9    =    4    +    5    +    0
10   =    9    +    1    +    0

• I tested it using computer, but it is impossible to write all the test result here. Anybody who know programming can test it to. Commented Aug 15, 2016 at 10:35
• arxiv.org/abs/0905.0635v3 Commented Aug 15, 2016 at 18:50
• @MichaelStocker: They are widely used in layman language, yes, but see en.wikipedia.org/wiki/Decimal_mark#Digit_grouping, and also most modern programming languages use "." for only the decimal point. It is really a pity people up to today cannot come to consensus on these basic important notations. Ever heard of bbc.com/news/magazine-27509559? Commented Aug 16, 2016 at 4:27
• @user21820 I suppose so. It is not difficult to prove these things when we do not require that the variables be non-negative, I did that much at the earlier question. In brief, positivity of variables is not a natural condition for quadratic forms. It is a natural condition for theta series of various types, so modular forms work. This means it may be possible to prove these conjectures for sufficiently large numbers, without being able to prove it for small numbers. Commented Aug 16, 2016 at 4:44
• @user21820 Oh, I do agree wholeheartedly. This should be standardized, as should many other things. But it hasn't yet. The closest we probably came to is: "The 22nd General Conference on Weights and Measures declared in 2003 that "the symbol for the decimal marker shall be either the point on the line or the comma on the line"" Commented Aug 16, 2016 at 9:44