Any larger pairs after this?

The most of credit for this solution goes to @Avishek Yadav as he aided me to get the idea of discriminant $D$ must be equal to $3Y^2$ $\begin{aligned} & 1+2+\cdots + M = 1^2+2^2+\cdots + N^2 \\& \frac{M(M+1)}{2} = \frac{N(N+1)(2N+1)}{6} \\& 3M^2 +3M -N(N+1)(2N+1) = 0 \\ & M = \frac{-3 \pm\sqrt{3 (8N^3+12N^2 +4N+3)}}{6} \end{aligned}$ Now noting the discriminant of above equation $\begin{aligned} & D = 8N^3+ 12N^2 +4N +3 \\& = 8\left(N+\frac{1}{2}\right)^3- 2\left(N+\frac{1}{2}\right)+3 \\ & = X^3 -X +3 \phantom{cccccc} \text{let x} = 2\left(N+\frac{1}{2}\right)\end{aligned}$ It is vivid that discriminant $D$ in the above equation will be perfect square iff $\begin{aligned} & D = 3Y^2 \\ & X^3 -X+3 = 3Y^2 \cdots(a)\end{aligned}$ The solution of equations $a$ has been provided by Saburô Uchiyama and one can Google it as $\text{ Solution of a Diophantine equation by saburô Uchiyama}$

So the solution for $M,N$ can be interred as $(0,0)$ , $(1,1)$ $(10,5)$ , $(13,6)$ , $(645,85)$

C++

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#include <iostream>
using namespace std;

int main()
{
    int m;
    double rn, rm;
    m = 1000;
    int i, n;
    for(i = 1; i<m; i++)
    {
        for(n = 1; n<i; n++)
        {
            rn = n*(n+1)*(2*n+1);
            rm = 3*i*(i+1);
            if(rn==rm)
            {
                cout << i << " " << n << "\n";
            }
        }
    }
}

I don't know how to use Python in Brilliant Solution -

Here is the python code -

M = 0

N = 0

for i in range (1000):

     M += i

     N = 0

     for j in range (M+1):

            N += (j*j)

            if N == M:

                  print (N + M)

Output = $\boxed{730}$

We get the next pair as - $( 645 , 85 )$

Python brute force

MATLAB code:

I found it easier to use $m$ as the driver. The sum of consecutive integers is $\frac{1}{2} n (n+1)$ and the sum of squares of consecutive integers is $1/6 m (1 + m) (1 + 2 m)$ . Solving for $n$ in terms of $m$ gives $\frac{1}{2} \left(\frac{\sqrt{8 m^3+12 m^2+4 m+3}}{\sqrt{3}}-1\right)$ . I then created a table of when that expression returned an integer: {{1,1},{10,5},{13,6},{645,85}}.