Organizing and displaying list-like result

Question

This is most likely a piece of cake for you guys, but it is difficult for me.

Related to a math problem, I examine if Fibonacci[n]*Fibonacci[n] + <some number K between 1 and 99> is a prime, and all this for first 1000 Fibonacci numbers only.

For example, this code: (K=1)

Select[Table[Fibonacci[n], {n, 1, 1000}], PrimeQ[#*# + 1] &]

returns

{1, 1, 2}

This code: (K=2)

Select[Table[Fibonacci[n], {n, 1, 1000}], PrimeQ[#*# + 2] &]

returns

{1, 1, 3, 21, 6765, 32951280099, \ 971183874599339129547649988289594072811608739584170445, \ 1082459262056433063877940200966638133809015267665311237542082678938909\ }

This code: (K=3)

Select[Table[Fibonacci[n], {n, 1, 1000}], PrimeQ[#*# + 3] &]

returns

{2, 8, 3524578, 27777890035288, \ 2011595611835338993891308327102733537615455242513357158345612749706882\ 9146295425939723629305572732574726246290673965789878845363842331040064\ 16432124798818261534841714338, \ 2949592466076064248964701302014885591673737506156850406413751530665307\ 5810241060939483954895520932111023343610904846943097162533007651451709\ 723277579925520157875345780869307228929160}

And this code: (K=99)

Select[Table[Fibonacci[n], {n, 1, 1000}], PrimeQ[#*# + 99] &]

returns

{2, 8, 3524578, 6557470319842, \ 4286863412788815942499567477797350205106309231244244822408841055026686\ 7672}

I need to get all results from 1 to 99, then count elements of each answer, and display these in the following (or similar) form:

   K    number of primes
   1            3
   2            8
   3            6
   .            .
   .            .
  99            5

I appreciate your help.

Answer 1

fibs2 = Fibonacci@Range@1000^2;
tab = Table[{k, Count[fibs2 + k, _?PrimeQ]}, {k, 1, 99}];

TableForm[tab, TableHeadings -> {None, {"K", "number of primes"}}]

(welcome back)