Primes and Expanded Pythagorean Triples of the form (a+b)(a-b)p-1=cp

 In every Pythagorean triple one of the three triples is always divisible by 5 (p=2, 2p+1=5). And, of course, 5 is prime.

Similarly, the proposition for the "expanded" Pythagorean triples is, if one of the three triples is always divisible by 2p+1, then 2p+1 is prime. However, if one of them is not always divisible by 2p+1, then 2p+1 is not prime.

You can determine many integer solutions for (a+b)(a-b)p-1=cp in the following manner:

Let v, u be positive integers such that v>u

Then c=v+u

If Mod(c,v-u) is 0, then a & b have integer solutions such that

a=(v+u)p+(v-u)p/2(v-u)(p-1)

b=a-(v-u)

Below are some examples.

Unfortunately, while Mod(c,v-u) provides many integer solutions, it does not provide all of the integer solutions. Perhaps it is possible to rewrite the equations in some manner similar to the solution for Pythagorean triples (p=2, v^2-u^2, 2uv, v^2+u^2).

If you would like more examples (for c up to 69 {when v-u=1} and p up to 1000), someone was nice enough to run the following Mathematica code that I turned into this file -- Warning! Warning! Warning! It's a relatively large file at 4+ Mb:

http://mysite.verizon.net/reriker/10003n.htm

Mathematica code:
Table[
    Table[a = Floor[c^p/2] + 1; b = a - 1;
       {p, 2p+1, c, Mod[a, 2p+1], Mod[b, 2p+1], Mod[c, 2p+1]},
       {c, 3, 69, 2}
    ],
    {p, 2, 1000}
]// MatrixForm

p     c   c^p a b 2p+1 MOD(a,2p+1) MOD(b,2p+1) MOD(c,2p+1)
  v u v+u MOD(c,v-u)   (v+u)^p+(v-u)^p/2(v-u)^(p-1)          
              a-(v-u)        
2 2 1 3 0 9 5 4 5 0 4 3
3 2 5 0 25 13 12 5 3 2 0
4 3 7 0 49 25 24 5 0 4 2
3 1 4 0 16 5 3 5 0 3 4
4 2 6 0 36 10 8 5 0 3 1
5 3 8 0 64 17 15 5 2 0 3
6 3 9 0 81 15 12 5 0 2 4
9 6 15 0 225 39 36 5 4 1 0
12 9 21 0 441 75 72 5 0 2 1
                       
3 2 1 3 0 27 14 13 7 0 6 3
3 2 5 0 125 63 62 7 0 6 5
4 3 7 0 343 172 171 7 4 3 0
3 1 4 0 64 9 7 7 2 0 4
4 2 6 0 216 28 26 7 0 5 6
5 3 8 0 512 65 63 7 2 0 1
6 3 9 0 729 42 39 7 0 4 2
9 6 15 0 3375 189 186 7 0 4 1
12 9 21 0 9261 516 513 7 5 2 0
                       
4 2 1 3 0 81 41 40 9 5 4 3
3 2 5 0 625 313 312 9 7 6 5
4 3 7 0 2401 1201 1200 9 4 3 7
3 1 4 0 256 17 15 9 8 6 4
4 2 6 0 1296 82 80 9 1 8 6
5 3 8 0 4096 257 255 9 5 3 8
6 3 9 0 6561 123 120 9 6 3 0
9 6 15 0 50625 939 936 9 3 0 6
12 9 21 0 194481 3603 3600 9 3 0 3
                       
5 2 1 3 0 243 122 121 11 1 0 3
3 2 5 0 3125 1563 1562 11 1 0 5
4 3 7 0 16807 8404 8403 11 0 10 7
3 1 4 0 1024 33 31 11 0 9 4
4 2 6 0 7776 244 242 11 2 0 6
5 3 8 0 32768 1025 1023 11 2 0 8
6 3 9 0 59049 366 363 11 3 0 9
9 6 15 0 759375 4689 4686 11 3 0 4
12 9 21 0 4084101 25212 25209 11 0 8 10
                       
6 2 1 3 0 729 365 364 13 1 0 3
3 2 5 0 15625 7813 7812 13 0 12 5
4 3 7 0 117649 58825 58824 13 0 12 7
3 1 4 0 4096 65 63 13 0 11 4
4 2 6 0 46656 730 728 13 2 0 6
5 3 8 0 262144 4097 4095 13 2 0 8
6 3 9 0 531441 1095 1092 13 3 0 9
9 6 15 0 11390625 23439 23436 13 0 10 2
12 9 21 0 85766121 176475 176472 13 0 10 8
                       
7 2 1 3 0 2187 1094 1093 15 14 13 3
3 2 5 0 78125 39063 39062 15 3 2 5
4 3 7 0 823543 411772 411771 15 7 6 7
3 1 4 0 16384 129 127 15 9 7 4
4 2 6 0 279936 2188 2186 15 13 11 6
5 3 8 0 2097152 16385 16383 15 5 3 8
6 3 9 0 4782969 3282 3279 15 12 9 9
9 6 15 0 170859375 117189 117186 15 9 6 0
12 9 21 0 1801088541 1235316 1235313 15 6 3 6
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