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## Main Question or Discussion Point

**Professional Help Needed--Elementary Number Theory**

I will preface this by saying that I have no formal training in Mathematics. I've taken Calc 1 and a couple of Symbolic Logic classes. Forgive me if I butchered any terminology. However, this has been bugging me for a while, and I would like to know if anyone knows anything about this.

There seems to be a predictable pattern in the gaps between successive terms in series with prime exponents. This has been known for x^2 for a long time.

x^1== +1

x^2 == 2x+1

x^3 == [(x^2+x)(3)] +1

1+ (1^2+1) (3) +1 = 8

8 + (2^2+2) (3) +1 = 27

27 + (3^2+3) (3) +1= 64 ...

x^5== [(x^2+x)(x^2+x+1)(5)]+1

1+[(1^2+1)(1^2+1+1)(5)]+1=32

32 + (2^2+2)(2^2+2+1)(5)+1=243

243+(3^2+3)(3^2+3+1)(5)+1=1024...

x^7 = {(x^2 + x) (x^2 + x + 1)^2 (7)} + 1

x^11 = {(x^2 + x) (x^2 + x +1) ((x^2 + x)^2 + (x^2 + x + 1)^3) (11)} + 1

x^13 = {(x^2 + x) (x^2 + x + 1)^2 (2 (x^2 + x)^2 + (x^2 + x + 1)^3) (13)} + 1

I have Mathematica notebooks with the computations for x^7,x^11, and x^13, verifying this up to x=100 and each series gap between successive members follows the pattern. However at x^17, I ran out of steam and cognitive resources. Prime factoring the [[gap term] -1] is how I started looking at the other series, and there is some frustratingly periodic behavior in the prime factors. Twin prime exponents are also connected in some way, but I cannot figure it out.

I have no idea what this means, if anything at all.