What Prime is the Next Minimal Sum of Squares?
All integers can be de-composed into a list of squares which sum to equal the initial integer. The minimal list of squares is derived sequentially by subtracting the largest square repeatedly. An example best illustrates (followed by Mathematica code to generate the Minimal Sum of Squares): 1973 = {44, 6, 1} 1936 = 44*44 36 = 6*6 1 = 1*1 Given this definition we now restrict the problem to the longest sequences produced by the smallest numbers, and we start to notice something very interesting 2 = {1,1} 3 = {1,1,1} 7 = {2,1,1,1} 23 = {4,2,1,1,1} 167 = {12,4,2,1,1,1} which are all prime. Q1. Are there any more primes in this sequence? Contributions came from Loris Cappelletti & Johann Wiesenbauer: Cappelletti wrote: If we consider the succession N0=1, N1=2, N2=3, N3=7, N4=23, N5=167, its not difficult to show that the smallest number producing a sequence of length k+2 is: Nk+1 = ((Nk + 1)/2)2 + Nk , for k=>2 With this formula, I found the following terms of the succession up to N15, and
