$($Due January $29)$ Prove that the function computing the product

$(1^2+1)**(2^2+1)**(3^2+1)**dots**(n^2+1)$

is primitive recursive. This proof should follow the same pattern that we used in class to prove that addition and multiplication are primitive recursive:

You start with a $3-$dot expression

First you write a for$-$loop corresponding to this function

Then you describe this for$-$loop in mathematical terms

Then, to prepare for a match with the general expression for primitive recursion, you rename the function to $f$ and the parameters to $n_1,$dots,$m$

Then you write down the general expression of primitive recursion for the corresponding $k$

Then you match: find $g$ and $h$ for which the specific case of primitive recursion will be exactly the functions corresponding to initialization and to what is happening inside the loop

Then, you get a final expression for the function

$(1^2+1)**(2^2+1)**(3^2+1)**dots**(n^2+1)$

that proves that this function is primitive recursive, i$.$e$.,$ that it can be formed from $0,{}_i^k,$ and $$ by composition and primitive recursion.