<s>[INST] Are these the only "intersections" between the following series, and why all of them are multiples of $10$ As a web developer that programs in PhP, I enjoy running some scripts to see some of math wonders, however, PhP is limited to large calculations.
I wanted to see if there are any intersections for the following $2$ series:
A) $1 + 2 + 3+ 4 + 5 + \ldots$
B) $(1) + (1 + 2) + (1 + 2 + 3) +(1 + 2 + 3 + 4) + (1 + 2 + 3 + 4 + 5) + \ldots$
Out of the first $116410911$ positive integers, I have "found" the following "intersections" between A) and B):
At the number $10$:
$10 - (1+2+3+4)=0$
$10 - ((1)+(1+2)+(1+2+3))=0$
At the number $120$:
$120 - (1+2+3+4+5+6+7+8+9+10+11+12+13+14+15) = 0$
$120 - ((1)+(1+2)+(1+2+3)+(1+2+3+4)+(1+2+3+4+5)+(1+2+3+4+5+6)+(1+2+3+4+5+6+7)+(1+2+3+4+5+6+7+8)) = 0$
And I am not going to list the other procedures, but also:
At the numbers $1540$ and $7140$.
Can anyone confirm if there are any more "intersections"? In other words, am I to expect a finite case or an infinite case? Also, is it a coincidence that all four of them are numbers that are multiples of $10$?
 [/INST] These are the numbers that are both "triangular" ($n(n+1)/2$) and "tetrahedral" ($n(n+1)(n+2)/6$). They are tabulated at https://oeis.org/A027568. You have found all of them. There are several links to the literature at the OEIS site. It's unlikely that there is a wholly elementary proof, as elliptic curves are involved.
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