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5 Clarified statement.

Well, if we consider n consecutive 4th powers with initial a,

F(a,n) = a^4 + (a+1)^4 + (a+2)^4 + ... + (a+n-1)^4

or, equivalently,

F(a,n) = (n/30)(-1+30a^2-60a^3+30a^4+30a(1-3a+2a^2)n+10(1-6a+6a^2)n^2+(-15+30a)n^3+6n^4)

it is easy to check that F(a,n) = y^4 (or even just y^2) has NO solution in the positive integers with BOTH {a or n,*n*} < 1000, with the exception of the trivial n = 1. (I had checked this with Mathematica some time back.)

If we relax your question and allow n 4th powers in arithmetic progression d equal to some kth power, then the smallest I found was 64 4th powers with common difference d = 2 starting with,

29^4 + 31^4 + 33^4 + ... + 155^4 = 96104^2

P.S. The closed-form formula for general d is available, but I find it too tedious to include in this post.

4 Last correction <sigh>

Well, if we consider n consecutive 4th powers with initial a,

F(a,n) = a^4 + (a+1)^4 + (a+2)^4 + ... + (a+n-1)^4

or, equivalently,

F(a,n) = (n/30)(-1+30a^2-60a^3+30a^4+30a(1-3a+2a^2)n+10(1-6a+6a^2)n^2+(-15+30a)n^3+6n^4)

it is easy to check that F(a,n) = y^4 (or even just y^2) has NO solution in the positive integers with BOTH a or n < 1000, with the exception of the trivial n = 1. (I checked.had checked this with Mathematica some time back.)

If we relax your question and allow n 4th powers in arithmetic progression d equal to some kth power, then the smallest I found was 64 4th powers with common difference d = 2 starting with,

29^4 + 31^4 + 33^4 + ... + 155^4 = 96104^2

P.S. The closed-form formula for general d is available, but I find it too tedious to include in this post.

3 Added closed-form formula for d = 1

Well, if we consider n consecutive 4th powers with initial a,

F(a,n) = a^4 + (a+1)^4 + (a+2)^4 + ... + (a+n-1)^4

or, equivalently,

F(a,n) = (n/30)(-1+30a^2-60a^3+30a^4+30a(1-3a+2a^2)n+10(1-6a+6a^2)n^2+(-15+30a)n^3+6n^4)

it is easy to check that F(a,n) = y^4 (or even just y^2) has NO solution in the positive integers with BOTH a or n < 1000. (I checked.)

If we relax your question and allow n 4th powers in arithmetic progression d equal to some kth power, then the smallest I found was 64 4th powers with common difference d = 2 starting with,

29^4 + 31^4 + 33^4 + ... + 155^4 = 96104^2

P.S. The closed-form formula for general d is available, but I find it too tedious to include in this post.

2 Clarified meaning.
1