$a^{3} + (a+1)^{3} +(a+2)^{3} +(a+3)^{3}+(a+4)^{3}+(a+5)^{3}+(a+6)^{3}= b^{4} +(b+1)^{4}$

What is the number of solutions $(a,b)$ satisfying the equation above?

The answer is 0.

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In the case of the LHS we can try modulo 7 and can write it that it is divisible by 7.since LHS can be expressed as 0(mod 7)

Now for RHS it can never be expressed as 0(mod 7) and hence no solutions exist