TITLE: On $1^2+2^2+\dots+24^2 = 70^2$, and $15^3+16^3+\dots+34^3 = 70^3$
QUESTION [13 upvotes]: It is quite well-known that,
$$1^2+2^2+\dots+24^2 = 70^2$$
Not so well-known is,
$$15^3+16^3+\dots+34^3 = 70^3$$
The formula for the sum of $m$ consecutive squares starting with $a^2$ is,
$$F(a,m) = (m/6)(6a^2-6a+6am+1-3m+2m^2)$$
while the sum of $n$ consecutive cubes starting with $b^3$ is,
$$F(b,n) = (n/4)(2b+n-1)(2b^2-2b+2bn-n+n^2)$$
Question: Is the only solution in positive integers to the simultaneous equations,
$$F(a,m) = x^2$$
$$F(b,n) = x^3$$
given by $a,m;b,n;x = 1,\,24;\,15,\,20;\,70$? (I have searched within a relatively small range, but didn't find any new solution.)

REPLY [9 votes]: Just as a wide comment.
My search limitation was $x<2 \times 10^6$.
There is only  $x=70$ in this range.
The set of such $x$, which can be written as sum of consequent cubes, is rather pure: here is list:
\begin{array}{rlr}
6^3 = & 3^3+4^3+5^3 & (3 ~terms) \\
20^3 = &  11^3 + ... + 14^3 & (4= \color{darkviolet}{2^2}~terms) \\ 
40^3 = &  3^3 + ... + 22^3 & (20 ~terms) \\
60^3 = & 6^3 + ... + 30^3 & (25 = \color{darkviolet}{5^2}~terms) \\
70^3 = & 15^3 + ... + 34^3 & (20 ~terms) \\
180^3 = & 6^3 + ... + 69^3 & (64 = \color{darkviolet}{8^2}=  \color{red}{4^3} ~terms) \\
330^3 = & 11^3 + ... + 109^3 & (99 ~terms) \\
540^3 = & 34^3 + ... + 158^3 & (125 = \color{red}{5^3} ~terms) \\
1155^3 = & 291^3 + ... + 339^3 & (49 = \color{darkviolet}{7^2} ~terms) \\
1581^3 = & 213^3 + ... + 365^3 & (153 ~terms) \\
2805^3 = & 556^3 + ... + 654^3 & (99 ~terms) \\
2856^3 = & 213^3 + ... + 555^3 & (343 = \color{red}{7^3} ~terms) \\
2856^3 = & 273^3 + ... + 560^3 & (288 ~terms) \\
3876^3 = & 646^3 + ... + 798^3 & (153 ~terms) \\
5544^3 = & 406^3 + ... + 917^3 & (512  = \color{red}{8^3} ~terms) \\
16830^3 = & 1134^3 + ... + 2133^3 & (1000  = \color{red}{10^3}~terms) \\
27060^3 = & 1735^3 + ... + 3065^3 & (1331  = \color{red}{11^3}~terms) \\
62244^3 = & 3606^3 + ... + 5802^3 & (2197 = \color{red}{13^3}~terms) \\
82680^3 = & 305^3 + ... + 6895^3 & (6591 ~terms) \\
90090^3 = & 4966^3 + ... + 7709^3 & (2744 = \color{red}{14^3}~terms) \\
175440^3 = & 8790^3 + ... + 12885^3 & (4096 = \color{darkviolet}{64^2} = \color{red}{16^3}~terms) \\
237456^3 = & 11368^3 + ... + 16280^3 & (4913 = \color{red}{17^3}~terms) \\
249424^3 = & 1624^3 + ... + 15784^3 & (14161 = \color{darkviolet}{119^2}~terms) \\
273819^3 = & 3010^3 + ... + 16932^3 & (13923 ~terms) \\
413820^3 = & 18171^3 + ... + 25029^3 & (6859 = \color{red}{19^3}~terms) \\
431548^3 = & 34228^3 + ... + 36076^3 & (1849 = \color{darkviolet}{43^2}~terms) \\
534660^3 = & 22534^3 + ... + 30533^3 & (8000 = \color{red}{20^3}~terms) \\
860706^3 = & 33558^3 + ... + 44205^3 & (10648 = \color{red}{22^3}~terms) \\
1074744^3 = & 40381^3 + ... + 52547^3  & (12167 = \color{red}{23^3} ~terms) \\
1205750^3 = & 18551^3 + ... + 51674^3 & (33124 = \color{darkviolet}{182^2}~terms)  \\       
1306620^3 = & 4880^3 + ... + 54655^3 & (49776 ~terms)   \\      
1630200^3 = & 57084^3 + ... + 72708^3 & (15625 = \color{darkviolet}{125^2} = \color{red}{25^3}  ~terms)  \\     
1764070^3 = & 46690^3 + ... + 71890^3 &  (25201 ~terms) \\    
1962820^3 = & 11170^3 + ... + 74170^3 &(63001 = \color{darkviolet}{251^2}~terms)   \\
1983150^3 = & 67150^3 + ... + 84725^3 & (17576 = \color{red}{26^3}  ~terms)  \\
\end{array}
And a few curious identities of this search:
power $3$:
$$
215^3+...+555^3 = {\Large{2856^3}} = 273^3+...+ 560^3
$$
power $2$:
$$
7^2+...+39^2 = {\Large{143^2}} = 38^2+...+ 48^2
$$
$$
294^2+...+367^2 = {\Large{2849^2}} = 854^2+...+ 864^2
$$
$$
2175^2+...+5199^2 = {\Large{208395^2}} = 29447^2+...+ 29496^2
$$
$$
9401^2+...+25273^2 = {\Large{2259257^2}} = 26181^2+...+ 32158^2
$$