Progressive Terms

Relevant wiki: Arithmetic Progressions

Let the number of terms be $n$ and the common difference be $d$

We know that $T_n = 2016$

$1729 + (n-1)(d) = 2016\\ (n-1)(d) = 287\\ (n-1)(d) = 1 \times 7 \times 41$

Since both $(n-1)$ and $d$ are positive integers, we have these possible combinations:

$n-1=1,\quad d= 7 \times 41\\ n-1=7,\quad d= 41\\ n-1=41,\quad d= 7\\ n-1=7 \times 41,\quad d=1$

Therefore, there are $\boxed{4}$ possible arithmetic progressions

The answer is the number of divisors of the differenze beetwen 2016 and 1729 :)

This is equivalent to solving the equation $ab=287$ over the positive integers.

The answer is hence 4.

We can write this like 1729 + N x D = 2016, where D is the common difference and N is the number of elements up to 2016 (excluding 1729)

So, N x D = 287.

We observe that 287 = 7 x 41.

Both 7 and 41 are prime.

So all possible combination for (N,D) are (1,287), (287,1), (7,41), (41,7).