Pseudo Pythagorean Triples (^p)

 The integer solutions -- the Pythagorean triples -- to a2+b2=c2 can be determined when u and v (v>u) are integers, such that a, b, c are v2-u2, 2uv, v2+u2 respectively.

Now let p also be an integer > 1, and let

a=vp+up, b=vp-up, c=2uv

then, for every u, v, and p (v>u):

{(a+b)[(a-b)(p-1)]}/{u[p(p-2)]}=cp

Although I am not going to try and show it here, if I have done it correctly, algebraically the left side of the equation does equal the right side. Below are some numerical examples.

Of note, similar to the Primes and Expanded Pythagorean Triples of the form (a+b)(a-b)p-1=cp, it appears that, 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.

Of course, even if this holds true, it may not be efficient enough to be of any value.

      v^p+u^p v^p-u^p 2uv                  
p v u a b c (a+b) (a-b)^(p-1) (a+b)(a-b)^(p-1) (a+b)[(a-b)^(p-1)]/u^(p(p-2)) c^p 2p+1 mod(a,2p+1) mod(b,2p+1) mod(c,2p+1)
2 2 1 5 3 4 8 2 16 16 16 5 0 3 4
2 3 1 10 8 6 18 2 36 36 36 5 0 3 1
2 4 1 17 15 8 32 2 64 64 64 5 2 0 3
2 5 1 26 24 10 50 2 100 100 100 5 1 4 0
2 3 2 13 5 12 18 8 144 144 144 5 3 0 2
2 4 2 20 12 16 32 8 256 256 256 5 0 2 1
2 5 2 29 21 20 50 8 400 400 400 5 4 1 0
2 4 3 25 7 24 32 18 576 576 576 5 0 2 4
2 5 3 34 16 30 50 18 900 900 900 5 4 1 0
2 6 3 45 27 36 72 18 1296 1296 1296 5 0 2 1
                             
3 2 1 9 7 4 16 4 64 64 64 7 2 0 4
3 3 1 28 26 6 54 4 216 216 216 7 0 5 6
3 4 1 65 63 8 128 4 512 512 512 7 2 0 1
3 5 1 126 124 10 250 4 1000 1000 1000 7 0 5 3
3 3 2 35 19 12 54 256 13824 1728 1728 7 0 5 5
3 4 2 72 56 16 128 256 32768 4096 4096 7 2 0 2
3 5 2 133 117 20 250 256 64000 8000 8000 7 0 5 6
3 4 3 91 37 24 128 2916 373248 13824 13824 7 0 2 3
3 5 3 152 98 30 250 2916 729000 27000 27000 7 5 0 2
3 6 3 243 189 36 432 2916 1259712 46656 46656 7 5 0 1
                             
4 2 1 17 15 4 32 8 256 256 256 9 8 6 4
4 3 1 82 80 6 162 8 1296 1296 1296 9 1 8 6
4 4 1 257 255 8 512 8 4096 4096 4096 9 5 3 8
4 5 1 626 624 10 1250 8 10000 10000 10000 9 5 3 1
4 3 2 97 65 12 162 32768 5308416 20736 20736 9 7 2 3
4 4 2 272 240 16 512 32768 16777216 65536 65536 9 2 6 7
4 5 2 641 609 20 1250 32768 40960000 160000 160000 9 2 6 2
4 4 3 337 175 24 512 4251528 2176782336 331776 331776 9 4 4 6
4 5 3 706 544 30 1250 4251528 5314410000 810000 810000 9 4 4 3
4 6 3 1377 1215 36 2592 4251528 11019960576 1679616 1679616 9 0 0 0
                             
5 2 1 33 31 4 64 16 1024 1024 1024 11 0 9 4
5 3 1 244 242 6 486 16 7776 7776 7776 11 2 0 6
5 4 1 1025 1023 8 2048 16 32768 32768 32768 11 2 0 8
5 5 1 3126 3124 10 6250 16 100000 100000 100000 11 2 0 10
5 3 2 275 211 12 486 16777216 8153726976 248832 248832 11 0 2 1
5 4 2 1056 992 16 2048 16777216 34359738368 1048576 1048576 11 0 2 5
5 5 2 3157 3093 20 6250 16777216 104857600000 3200000 3200000 11 0 2 9
5 4 3 1267 781 24 2048 55788550416 114254951251968 7962624 7962624 11 2 0 2
5 5 3 3368 2882 30 6250 55788550416 348678440100000 24300000 24300000 11 2 0 8
5 6 3 8019 7533 36 15552 55788550416 867623536069632 60466176 60466176 11 0 9 3
                             
6 2 1 65 63 4 128 32 4096 4096 4096 13 0 11 4
6 3 1 730 728 6 1458 32 46656 46656 46656 13 2 0 6
6 4 1 4097 4095 8 8192 32 262144 262144 262144 13 2 0 8
6 5 1 15626 15624 10 31250 32 1000000 1000000 1000000 13 0 11 10
6 3 2 793 665 12 1458 34359738368 50096498540544 2985984 2985984 13 0 2 12
6 4 2 4160 4032 16 8192 34359738368 281474976710656 16777216 16777216 13 0 2 3
6 5 2 15689 15561 20 31250 34359738368 1.073741824E+15 64000000 64000000 13 11 0 7
6 4 3 4825 3367 24 8192 6.58851622702877E+15 5.39731249318197E+19 191102976 191102976 13 2 0 11
6 5 3 16354 14896 30 31250 6.58851622702877E+15 2.05891132094649E+20 729000000 729000000 13 0 11 4
6 6 3 47385 45927 36 93312 6.58851622702877E+15 6.14787626176509E+20 2176782336 2176782336 13 0 11 10
                             
7 2 1 129 127 4 256 64 16384 16384 16384 15 9 7 4
7 3 1 2188 2186 6 4374 64 279936 279936 279936 15 13 11 6
7 4 1 16385 16383 8 32768 64 2097152 2097152 2097152 15 5 3 8
7 5 1 78126 78124 10 156250 64 10000000 10000000 10000000 15 6 4 10
7 3 2 2315 2059 12 4374 281474976710656 1.23117154813241E+18 35831808 35831808 15 5 4 12
7 4 2 16512 16256 16 32768 281474976710656 9.22337203685478E+18 268435456 268435456 15 12 11 1
7 5 2 78253 77997 20 156250 281474976710656 4.398046511104E+19 1280000000 1280000000 15 13 12 5
7 4 3 18571 14197 24 32768 7.00281530441679E+21 2.29468251895129E+26 4586471424 4586471424 15 1 7 9
7 5 3 80312 75938 30 156250 7.00281530441679E+21 1.09418989131512E+27 21870000000 21870000000 15 2 8 0
7 6 3 282123 277749 36 559872 7.00281530441679E+21 3.92068021011444E+27 78364164096 78364164096 15 3 9 6
                             
8 2 1 257 255 4 512 128 65536 65536 65536 17 2 0 4
8 3 1 6562 6560 6 13122 128 1679616 1679616 1679616 17 0 15 6
8 4 1 65537 65535 8 131072 128 16777216 16777216 16777216 17 2 0 8
8 5 1 390626 390624 10 781250 128 100000000 100000000 100000000 17 0 15 10
8 3 2 6817 6305 12 13122 9.22337203685478E+18 1.21029087867608E+23 429981696 429981696 17 0 15 12
8 4 2 65792 65280 16 131072 9.22337203685478E+18 1.20892581961463E+24 4294967296 4294967296 17 2 0 16
8 5 2 390881 390369 20 781250 9.22337203685478E+18 7.20575940379279E+24 25600000000 25600000000 17 0 15 3
8 4 3 72097 58975 24 131072 6.69884970275022E+28 8.78031628238876E+33 110075314176 110075314176 17 0 2 7
8 5 3 397186 384064 30 781250 6.69884970275022E+28 5.23347633027361E+34 656100000000 656100000000 17 15 0 13
8 6 3 1686177 1673055 36 3359232 6.69884970275022E+28 2.2502990284669E+35 2821109907456 2821109907456 17 15 0 2
                             
9 2 1 513 511 4 1024 256 262144 262144 262144 19 0 17 4
9 3 1 19684 19682 6 39366 256 10077696 10077696 10077696 19 0 17 6
9 4 1 262145 262143 8 524288 256 134217728 134217728 134217728 19 2 0 8
9 5 1 1953126 1953124 10 3906250 256 1000000000 1000000000 1000000000 19 2 0 10
9 3 2 20195 19171 12 39366 1.20892581961463E+24 4.75905738149495E+28 5159780352 5159780352 19 17 0 12
9 4 2 262656 261632 16 524288 1.20892581961463E+24 6.33825300114115E+29 68719476736 68719476736 19 0 2 16
9 5 2 1953637 1952613 20 3906250 1.20892581961463E+24 4.72236648286965E+30 512000000000 512000000000 19 0 2 1
9 4 3 281827 242461 24 524288 5.76727028350443E+36 3.02371060239797E+42 2641807540224 2641807540224 19 0 2 5
9 5 3 1972808 1933442 30 3906250 5.76727028350443E+36 2.25283995449392E+43 19683000000000 19683000000000 19 0 2 11
9 6 3 10097379 10058013 36 20155392 5.76727028350443E+36 1.16241593333983E+44 101559956668416 101559956668416 19 0 2 17


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