Diophantine equation: x^2 + y^2 = z^2 + 1 (Almost Pythagorean Triples)

A Pythagorean triple (PT) is an ordered triple   (a, b, c)   that satisfies the Pythagorean equation:

$a^2 \; + \; b^2 \; = \; c^2$

Find an ordered triple   (x, y, z)   that satisfies the Diophantine equation

$x^2 \; + \; y^2 \; = \; z^2 \; + \; 1$

where   x,   y   and   z   are positive integers.

In the ” Concluding Remark” section:

However, the result in Theorem 2.2 does not generate all almost Pythagorean triples explicitly since we restrict there our PPTs to be of the form:

$(a, b, c) \; = \; ( \,2 \,i - 1, \; 2 \,i^2 - 2 \,i, \; 2 \,i^2 - 2 \,i + 1 \,)$

As a possible extension to this note, we encourage the readers to answer the problem:
“Is there an explicit formula that generates all almost Pythagorean triples?
If none, given two explicit solutions (comes in pair)   $s_1$   and   $s_2$ ,   can we possibly device a criterion that determines which one is ideal, perhaps in terms of the number of APT’s being generated?”.

math grad - Interest: Number theory
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7 Responses to Diophantine equation: x^2 + y^2 = z^2 + 1 (Almost Pythagorean Triples)

1. pipo says:

There are a lot:
Here all of x+y+z < 1000
Format: [x+y+z, x, y, z]
[19, 7, 4, 8]
[29, 9, 8, 12]
[31, 11, 7, 13]
[41, 13, 11, 17]
[41, 17, 6, 18]
[43, 15, 10, 18]
[49, 19, 9, 21]
[53, 17, 14, 22]
[55, 19, 13, 23]
[65, 21, 17, 27]
[67, 23, 16, 28]
[71, 26, 15, 30]
[71, 29, 11, 31]
[71, 31, 8, 32]
[77, 25, 20, 32]
[79, 27, 19, 33]
[79, 31, 14, 34]
[89, 29, 23, 37]
[89, 34, 17, 38]
[91, 31, 22, 38]
[97, 41, 13, 43]
[101, 33, 26, 42]
[101, 41, 16, 44]
[103, 35, 25, 43]
[109, 43, 19, 47]
[109, 49, 10, 50]
[111, 34, 31, 46]
[111, 41, 23, 47]
[113, 37, 29, 47]
[115, 39, 28, 48]
[125, 41, 32, 52]
[127, 43, 31, 53]
[127, 55, 15, 57]
[129, 49, 25, 55]
[131, 53, 21, 57]
[137, 45, 35, 57]
[139, 47, 34, 58]
[139, 55, 24, 60]
[149, 49, 38, 62]
[151, 51, 37, 63]
[151, 56, 31, 64]
[153, 65, 20, 68]
[155, 64, 23, 68]
[155, 71, 12, 72]
[161, 53, 41, 67]
[161, 65, 26, 70]
[161, 71, 17, 73]
[163, 55, 40, 68]
[169, 50, 49, 70]
[169, 64, 33, 72]
[169, 67, 29, 73]
[173, 57, 44, 72]
[175, 59, 43, 73]
[181, 55, 51, 75]
[181, 64, 41, 76]
[181, 76, 25, 80]
[183, 79, 22, 82]
[185, 61, 47, 77]
[187, 63, 46, 78]
[191, 71, 39, 81]
[191, 77, 31, 83]
[197, 65, 50, 82]
[197, 71, 43, 83]
[199, 67, 49, 83]
[199, 79, 34, 86]
[199, 89, 19, 91]
[209, 69, 53, 87]
[209, 79, 41, 89]
[209, 89, 27, 93]
[209, 97, 14, 98]
[211, 71, 52, 88]
[221, 73, 56, 92]
[221, 89, 36, 96]
[223, 75, 55, 93]
[229, 91, 39, 99]
[231, 86, 47, 98]
[233, 77, 59, 97]
[235, 79, 58, 98]
[239, 71, 69, 99]
[239, 99, 35, 105]
[239, 103, 29, 107]
[241, 76, 65, 100]
[241, 109, 21, 111]
[245, 81, 62, 102]
[247, 83, 61, 103]
[249, 94, 49, 106]
[251, 76, 71, 104]
[251, 101, 41, 109]
[251, 111, 26, 114]
[257, 85, 65, 107]
[259, 87, 64, 108]
[259, 103, 44, 112]
[265, 97, 56, 112]
[265, 111, 37, 117]
[265, 113, 34, 118]
[269, 89, 68, 112]
[271, 91, 67, 113]
[271, 101, 55, 115]
[271, 118, 31, 122]
[271, 127, 16, 128]
[281, 93, 71, 117]
[281, 113, 46, 122]
[283, 95, 70, 118]
[287, 89, 79, 119]
[287, 111, 53, 123]
[287, 131, 23, 133]
[289, 109, 57, 123]
[289, 115, 49, 125]
[289, 129, 28, 132]
[293, 97, 74, 122]
[295, 99, 73, 123]
[295, 127, 36, 132]
[305, 101, 77, 127]
[305, 134, 33, 138]
[307, 103, 76, 128]
[307, 109, 69, 129]
[307, 120, 55, 132]
[309, 92, 89, 128]
[311, 116, 63, 132]
[311, 125, 51, 135]
[317, 105, 80, 132]
[319, 107, 79, 133]
[319, 127, 54, 138]
[321, 97, 91, 133]
[321, 137, 41, 143]
[323, 116, 71, 136]
[323, 134, 47, 142]
[329, 109, 83, 137]
[329, 124, 65, 140]
[331, 111, 82, 138]
[337, 129, 64, 144]
[337, 155, 25, 157]
[339, 118, 79, 142]
[341, 113, 86, 142]
[341, 137, 56, 148]
[341, 151, 35, 155]
[341, 161, 18, 162]
[343, 115, 85, 143]
[349, 139, 59, 151]
[349, 146, 49, 154]
[351, 127, 76, 148]
[351, 131, 71, 149]
[351, 151, 43, 157]
[353, 117, 89, 147]
[355, 119, 88, 148]
[365, 121, 92, 152]
[367, 123, 91, 153]
[369, 139, 73, 157]
[371, 149, 61, 161]
[373, 169, 32, 172]
[377, 125, 95, 157]
[377, 161, 48, 168]
[379, 113, 109, 157]
[379, 127, 94, 158]
[379, 151, 64, 164]
[379, 169, 37, 173]
[389, 129, 98, 162]
[391, 118, 111, 162]
[391, 131, 97, 163]
[391, 146, 79, 166]
[391, 181, 27, 183]
[401, 133, 101, 167]
[401, 161, 66, 174]
[403, 135, 100, 168]
[407, 169, 59, 179]
[407, 175, 50, 182]
[409, 154, 81, 174]
[409, 163, 69, 177]
[413, 137, 104, 172]
[415, 127, 116, 172]
[415, 139, 103, 173]
[415, 181, 47, 187]
[417, 131, 113, 173]
[417, 170, 65, 182]
[419, 134, 111, 174]
[419, 153, 89, 177]
[419, 188, 39, 192]
[419, 191, 34, 194]
[419, 199, 20, 200]
[425, 141, 107, 177]
[427, 143, 106, 178]
[431, 161, 87, 183]
[431, 173, 71, 187]
[431, 191, 44, 196]
[433, 154, 97, 182]
[433, 181, 61, 191]
[433, 185, 55, 193]
[437, 145, 110, 182]
[439, 147, 109, 183]
[439, 175, 74, 190]
[441, 155, 101, 185]
[441, 181, 67, 193]
[449, 134, 129, 186]
[449, 149, 113, 187]
[449, 161, 99, 189]
[449, 169, 89, 191]
[449, 197, 49, 203]
[449, 209, 29, 211]
[451, 151, 112, 188]
[461, 139, 131, 191]
[461, 153, 116, 192]
[461, 185, 76, 200]
[461, 208, 41, 212]
[463, 144, 127, 192]
[463, 155, 115, 193]
[463, 199, 57, 207]
[469, 187, 79, 203]
[469, 209, 46, 214]
[471, 176, 95, 200]
[473, 157, 119, 197]
[475, 159, 118, 198]
[485, 161, 122, 202]
[485, 188, 89, 208]
[487, 163, 121, 203]
[489, 184, 97, 208]
[489, 209, 62, 218]
[491, 197, 81, 213]
[491, 204, 71, 216]
[495, 191, 92, 212]
[497, 165, 125, 207]
[499, 167, 124, 208]
[499, 199, 84, 216]
[505, 183, 109, 213]
[505, 197, 91, 217]
[505, 229, 43, 233]
[505, 241, 22, 242]
[509, 169, 128, 212]
[511, 171, 127, 213]
[511, 191, 103, 217]
[511, 239, 31, 241]
[517, 216, 73, 228]
[519, 155, 149, 215]
[519, 223, 64, 232]
[519, 239, 38, 242]
[521, 161, 144, 216]
[521, 173, 131, 217]
[521, 209, 86, 226]
[521, 231, 53, 237]
[523, 175, 130, 218]
[529, 199, 105, 225]
[529, 211, 89, 229]
[531, 160, 151, 220]
[533, 177, 134, 222]
[535, 179, 133, 223]
[545, 181, 137, 227]
[545, 188, 129, 228]
[545, 209, 103, 233]
[545, 233, 69, 243]
[547, 183, 136, 228]
[551, 206, 111, 234]
[551, 221, 91, 239]
[551, 251, 45, 255]
[557, 185, 140, 232]
[559, 187, 139, 233]
[559, 199, 125, 235]
[559, 223, 94, 242]
[559, 244, 63, 252]
[559, 249, 55, 255]
[569, 189, 143, 237]
[569, 214, 113, 242]
[571, 181, 153, 237]
[571, 191, 142, 238]
[571, 241, 77, 253]
[573, 209, 122, 242]
[573, 265, 40, 268]
[575, 206, 127, 242]
[575, 239, 83, 253]
[575, 247, 71, 257]
[577, 271, 33, 273]
[579, 239, 86, 254]
[581, 193, 146, 242]
[581, 233, 96, 252]
[583, 195, 145, 243]
[589, 176, 169, 244]
[589, 235, 99, 255]
[591, 221, 119, 251]
[593, 186, 161, 246]
[593, 197, 149, 247]
[593, 260, 65, 268]
[595, 199, 148, 248]
[599, 209, 139, 251]
[599, 254, 79, 266]
[599, 274, 47, 278]
[599, 287, 24, 288]
[601, 181, 171, 249]
[601, 251, 85, 265]
[601, 257, 76, 268]
[605, 201, 152, 252]
[607, 203, 151, 253]
[609, 229, 121, 259]
[611, 245, 101, 265]
[611, 271, 62, 278]
[617, 205, 155, 257]
[619, 207, 154, 258]
[619, 247, 104, 268]
[629, 209, 158, 262]
[631, 211, 157, 263]
[631, 236, 127, 268]
[631, 271, 78, 282]
[637, 289, 54, 294]
[639, 199, 175, 265]
[639, 287, 59, 293]
[641, 213, 161, 267]
[641, 257, 106, 278]
[643, 215, 160, 268]
[645, 208, 169, 268]
[645, 254, 113, 278]
[647, 191, 188, 268]
[647, 239, 134, 274]
[647, 305, 35, 307]
[649, 199, 181, 269]
[649, 244, 129, 276]
[649, 259, 109, 281]
[649, 289, 64, 296]
[649, 298, 49, 302]
[653, 217, 164, 272]
[655, 219, 163, 273]
[657, 281, 83, 293]
[659, 197, 189, 273]
[659, 239, 142, 278]
[659, 274, 95, 290]
[665, 221, 167, 277]
[667, 223, 166, 278]
[671, 202, 191, 278]
[671, 251, 135, 285]
[671, 269, 111, 291]
[677, 225, 170, 282]
[679, 227, 169, 283]
[679, 271, 114, 294]
[681, 241, 154, 286]
[681, 307, 61, 313]
[683, 265, 125, 293]
[683, 287, 94, 302]
[683, 311, 56, 316]
[685, 244, 153, 288]
[685, 286, 97, 302]
[687, 295, 85, 307]
[689, 229, 173, 287]
[689, 259, 137, 293]
[689, 321, 44, 324]
[691, 231, 172, 288]
[701, 233, 176, 292]
[701, 246, 161, 294]
[701, 251, 155, 295]
[701, 281, 116, 304]
[701, 311, 71, 319]
[701, 323, 51, 327]
[701, 337, 26, 338]
[703, 235, 175, 293]
[703, 271, 131, 301]
[703, 274, 127, 302]
[703, 287, 109, 307]
[703, 307, 79, 317]
[709, 283, 119, 307]
[711, 266, 143, 302]
[713, 237, 179, 297]
[713, 305, 90, 318]
[715, 239, 178, 298]
[721, 265, 151, 305]
[721, 341, 37, 343]
[725, 241, 182, 302]
[727, 243, 181, 303]
[727, 265, 155, 307]
[727, 298, 111, 318]
[729, 218, 209, 302]
[729, 274, 145, 310]
[731, 293, 121, 317]
[737, 245, 185, 307]
[737, 323, 81, 333]
[739, 247, 184, 308]
[739, 295, 124, 320]
[739, 329, 73, 337]
[741, 223, 211, 307]
[743, 309, 107, 327]
[743, 319, 92, 332]
[749, 239, 199, 311]
[749, 249, 188, 312]
[749, 324, 89, 336]
[751, 251, 187, 313]
[751, 281, 151, 319]
[751, 351, 46, 354]
[753, 289, 142, 322]
[755, 233, 209, 313]
[755, 287, 146, 322]
[755, 349, 53, 353]
[761, 253, 191, 317]
[761, 305, 126, 330]
[763, 255, 190, 318]
[769, 241, 209, 319]
[769, 289, 153, 327]
[769, 307, 129, 333]
[769, 321, 109, 339]
[769, 329, 97, 343]
[769, 349, 65, 355]
[773, 257, 194, 322]
[775, 259, 193, 323]
[781, 271, 183, 327]
[781, 321, 118, 342]
[781, 339, 91, 351]
[783, 307, 139, 337]
[785, 261, 197, 327]
[787, 263, 196, 328]
[791, 296, 159, 336]
[791, 317, 131, 343]
[791, 351, 80, 360]
[797, 265, 200, 332]
[799, 239, 229, 331]
[799, 267, 199, 333]
[799, 274, 191, 334]
[799, 319, 134, 346]
[799, 324, 127, 348]
[799, 343, 99, 357]
[799, 379, 39, 381]
[809, 269, 203, 337]
[809, 304, 161, 344]
[811, 244, 231, 336]
[811, 271, 202, 338]
[811, 289, 181, 341]
[811, 376, 55, 380]
[811, 391, 28, 392]
[813, 295, 175, 343]
[815, 254, 223, 338]
[815, 287, 186, 342]
[815, 371, 67, 377]
[821, 273, 206, 342]
[821, 329, 136, 356]
[823, 275, 205, 343]
[825, 353, 104, 368]
[827, 296, 183, 348]
[827, 344, 119, 364]
[829, 331, 139, 359]
[829, 369, 82, 378]
[831, 311, 167, 353]
[833, 277, 209, 347]
[835, 279, 208, 348]
[845, 281, 212, 352]
[847, 283, 211, 353]
[847, 370, 95, 382]
[849, 319, 169, 361]
[851, 341, 141, 369]
[853, 356, 121, 376]
[855, 367, 106, 382]
[857, 285, 215, 357]
[859, 287, 214, 358]
[859, 300, 199, 360]
[859, 343, 144, 372]
[859, 386, 79, 394]
[869, 260, 249, 360]
[869, 289, 218, 362]
[869, 404, 57, 408]
[871, 291, 217, 363]
[871, 326, 175, 370]
[881, 265, 251, 365]
[881, 293, 221, 367]
[881, 321, 188, 372]
[881, 342, 161, 378]
[881, 353, 146, 382]
[881, 377, 111, 393]
[881, 386, 97, 398]
[881, 391, 89, 401]
[881, 419, 41, 421]
[883, 271, 246, 366]
[883, 295, 220, 368]
[883, 337, 169, 377]
[883, 407, 64, 412]
[883, 415, 50, 418]
[889, 334, 177, 378]
[889, 355, 149, 385]
[893, 297, 224, 372]
[895, 299, 223, 373]
[901, 286, 241, 374]
[901, 351, 163, 387]
[901, 406, 81, 414]
[901, 409, 76, 416]
[905, 301, 227, 377]
[907, 303, 226, 378]
[911, 287, 246, 378]
[911, 341, 183, 387]
[911, 351, 170, 390]
[911, 365, 151, 395]
[911, 379, 131, 401]
[911, 391, 113, 407]
[911, 417, 71, 423]
[917, 305, 230, 382]
[919, 298, 239, 382]
[919, 307, 229, 383]
[919, 367, 154, 398]
[919, 379, 137, 403]
[919, 409, 91, 419]
[929, 309, 233, 387]
[929, 349, 185, 395]
[929, 433, 59, 437]
[929, 449, 30, 450]
[931, 311, 232, 388]
[937, 334, 209, 394]
[937, 391, 133, 413]
[937, 401, 118, 418]
[937, 433, 66, 438]
[939, 281, 269, 389]
[941, 313, 236, 392]
[941, 377, 156, 408]
[943, 315, 235, 393]
[945, 296, 257, 392]
[947, 431, 78, 438]
[949, 379, 159, 411]
[951, 286, 271, 394]
[951, 305, 251, 395]
[951, 356, 191, 404]
[951, 419, 101, 431]
[953, 317, 239, 397]
[953, 341, 211, 401]
[953, 449, 52, 452]
[955, 319, 238, 398]
[961, 337, 221, 403]
[961, 369, 181, 411]
[961, 406, 129, 426]
[961, 441, 73, 447]
[965, 321, 242, 402]
[967, 323, 241, 403]
[967, 351, 208, 408]
[967, 415, 120, 432]
[967, 461, 43, 463]
[969, 364, 193, 412]
[971, 389, 161, 421]
[971, 431, 98, 442]
[977, 325, 245, 407]
[979, 327, 244, 408]
[979, 391, 164, 424]
[985, 289, 288, 408]
[987, 341, 233, 413]
[987, 436, 103, 448]
[989, 305, 274, 410]
[989, 329, 248, 412]
[989, 362, 209, 418]
[989, 404, 153, 432]
[989, 417, 134, 438]
[989, 419, 131, 439]
[991, 309, 271, 411]
[991, 331, 247, 413]
[991, 371, 199, 421]
[991, 433, 111, 447]
[991, 463, 61, 467]
[993, 425, 125, 443]
[995, 414, 143, 438]

2. paul says:

Here are a few facts about these almost PT’s.
Assume our ordered triple (x, y, z) is x<y=6. (2)

For all Odd(x) the maximum y is (x^2-5)/4
For all even(x) the maximum y is (x^2-2)/2

For all odd(x) the differences between z and y (z-y) is always even.
For all even(x) the differences between z and y (z-y) is always odd.

For all odd(x) the maximum value of y gives (z-y) = 1.
For all even(x) the maximum value of y gives (z-y) = 2.

Proof of (2)

assuming Odd(x)

we can write x = 2n+1, n^2-5 = 4n^2+4n-4, which factors to 4(n^2+n-1) which is always divisible by 4.

assuming even(x)

we can write x = 2n, n^2-2 = 4n^2-2, which factors to 2(2n^2-1) which is always divisible by 2.

The limit of the minimum value of y is not so easy to find, nor is it easy to find how many solutions there are to any particular value of x. we can find them quite easily by solving equations that have integer values to
x^2+y^2=(y+m)^2 +1, where m is an integer value between Max y and
(x^2 – x)*Sqrt(x^4 – 2x^3 +x^2 – 1). With odd(m) if even(x) and visa versa.

here are a few examples when x = 500 to 504, the format is {x, {{y, z},{y, z},….}}

{500,{{665,832},{41665,41668},{124999,125000}}}

{501,{{1205,1305},{2485,2535},{6265,6285},{12545,12555},{31373,31377},{62749,62751}}}

{502,{{671,838},{41999,42002},{126001,126002}}}

{503,{{669,837},{941,1067},{1464,1548},{1721,1793},{2231,2287},{2991,3033},{3496,3532},{4504,4532},{5259,5283},{7019,7037},{9029,9043},{10536,10548},{15809,15817},{21081,21087},{31624,31628},{63251,63253}}}

Paul.

• paul says:

Don’t know how it managed to print this line

Assume our ordered triple (x, y, z) is x<y=6. (2)
Assume our ordered triple (x, y, z) is x<y=6. (2)

so first 4 lines should read

Here are a few facts about these almost PT’s.
Assume our ordered triple (x, y, z) is x<y=6. (2)

Paul.

3. benvitalis says:

Check my post

4. paul says:

Makes interesting reading however, I find it very limited in what their formulas generate, I acknowledge that they say it doesn’t find all APPT’s but their formulas only find 2 for any combination of {k and I} when there are in some cases many more, the examples given with small values doesn’t portray how many they miss, here is a small example using their formula iterated over {2 and 10} for {k and I} generates these

{1,2} , {4,7,8}, {5,5,7}
{1,3} , {6,17,18}, {9,19,21}
{1,4} , {8,31,32}, {13,41,43}
{1,5} , {10,49,50}, {17,71,73}
{1,6} , {12,71,72}, {21,109,111}
{1,7} , {14,97,98}, {25,155,157}
{1,8} , {16,127,128}, {29,209,211}
{1,9} , {18,161,162}, {33,271,273}
{1,10} , {20,199,200}, {37,341,343}
{2,2} , {7,11,13}, {8,9,12}
{2,3} , {11,29,31}, {14,31,34}
{2,4} , {15,55,57}, {20,65,68}
{2,5} , {19,89,91}, {26,111,114}
{2,6} , {23,131,133}, {32,169,172}
{2,7} , {27,181,183}, {38,239,242}
{2,8} , {31,239,241}, {44,321,324}
{2,9} , {35,305,307}, {50,415,418}
{2,10} , {39,379,381}, {56,521,524}

We note that the maximum x value in that list is 56 and using that value as the upper limit, the following are all APPT’s with an x value up to 56

{6,{{17,18}}}
{7,{{11,13}}}
{8,{{9,12},{31,32}}}
{9,{{19,21}}}
{10,{{15,18},{49,50}}}
{11,{{13,17},{29,31}}}
{12,{{71,72}}}
{13,{{19,23},{41,43}}}
{14,{{17,22},{31,34},{97,98}}}
{15,{{26,30},{55,57}}}
{16,{{23,28},{41,44},{127,128}}}
{17,{{21,27},{34,38},{71,73}}}
{18,{{161,162}}}
{19,{{27,33},{43,47},{89,91}}}
{20,{{25,32},{65,68},{199,200}}}
{21,{{53,57},{109,111}}}
{22,{{31,38},{79,82},{241,242}}}
{23,{{29,37},{41,47},{64,68},{131,133}}}
{24,{{55,60},{287,288}}}
{25,{{35,43},{49,55},{76,80},{155,157}}}
{26,{{33,42},{65,70},{111,114},{337,338}}}
{27,{{89,93},{181,183}}}
{28,{{39,48},{129,132},{391,392}}}
{29,{{37,47},{67,73},{103,107},{209,211}}}
{30,{{449,450}}}
{31,{{34,46},{43,53},{56,64},{77,83},{118,122},{239,241}}}
{32,{{41,52},{169,172},{511,512}}}
{33,{{64,72},{134,138},{271,273}}}
{34,{{47,58},{79,86},{113,118},{191,194},{577,578}}}
{35,{{45,57},{99,105},{151,155},{305,307}}}
{36,{{89,96},{127,132},{647,648}}}
{37,{{51,63},{111,117},{169,173},{341,343}}}
{38,{{49,62},{239,242},{721,722}}}
{39,{{71,81},{91,99},{188,192},{379,381}}}
{40,{{55,68},{265,268},{799,800}}}
{41,{{53,67},{64,76},{79,89},{101,109},{137,143},{208,212},{419,421}}}
{42,{{881,882}}}
{43,{{59,73},{71,83},{151,157},{229,233},{461,463}}}
{44,{{57,72},{103,112},{191,196},{321,324},{967,968}}}
{45,{{251,255},{505,507}}}
{46,{{63,78},{113,122},{209,214},{351,354},{1057,1058}}}
{47,{{61,77},{86,98},{134,142},{181,187},{274,278},{551,553}}}
{48,{{161,168},{1151,1152}}}
{49,{{50,70},{67,83},{94,106},{115,125},{146,154},{197,203},{298,302},{599,601}}}
{50,{{65,82},{175,182},{415,418},{1249,1250}}}
{51,{{55,75},{125,135},{323,327},{649,651}}}
{52,{{71,88},{449,452},{1351,1352}}}
{53,{{69,87},{111,123},{231,237},{349,353},{701,703}}}
{54,{{127,138},{289,294},{1457,1458}}}
{55,{{75,93},{101,115},{120,132},{185,193},{249,255},{376,380},{755,757}}}
{56,{{73,92},{97,112},{137,148},{311,316},{521,524},{1567,1568}}}

and using one of the larger examples they quote in the paper when x=155976 when they give their two examples here are how many there actually are

155976 with
{158327,222252}
{258103,301572}
{374345,405540}
{456247,482172}
{532169,554556}
{945055,957840}
{1321217,1330392}
{1946593,1952832}
{2343455,2348640}
{4755959,4758516}
{6628105,6629940}
{7975799,7977324}
{11729719,11730756}
{28621567,28621992}
{33144929,33145296}
{39882655,39882960}
{143108855,143108940}
{199414007,199414068}
{486570239,486570264}
{715544479,715544496}
{2432851255,2432851260}
{12164256287,12164256288}

I also find it unusual that they should make an OEIS entry with a sequence that is far from complete.
A final note, the published sequence in OEIS when k=1 and I>2 are all the even “x” values and the maximum “y” value for that particular “x” or it is simply {x, (x^2 – 2)/2, (x^2 – 2)/2 +1}. Filling in the blanks for odd “x” would be {x, (x^2 – 5)/4, (x^2 – 5)/4 +2}.

I’m still looking at a general algorithm to find all APPT’s.

Paul.

5. paul says:

Here is MMa code that will find all APPT’s with any given x value, (x<y<z). It could be made to fit into 1 line of code if it just printed the results as it found them, however this code builds up a table and prints the table at each iteration and sorts them as well as counting how many it finds within a range, so it's considerably longer than necessary, so here's the code

tot=0;Timing[m=6;
While[m<=50,If[OddQ[m],t=1,t=0];k={};
Do[If[IntegerQ[(m^2-((n+t)^2+1))/(2(n+t))],
AppendTo[k,{m,(m^2-((n+t)^2+1))/(2(n+t)),(m^2-((n+t)^2+1))/(2(n+t))+n}]],{n,1,Ceiling[-m+Sqrt[2m^2-1]]-2,2}];
Print[Reverse[k]];tot=tot+Length[k];m++];tot]

this results in this table

{{6,17,18}}
{{7,11,12}}
{{8,9,12},{8,31,32}}
{{9,8,11},{9,19,20}}
{{10,15,18},{10,49,50}}
{{11,13,16},{11,29,30}}
{{12,71,72}}
{{13,11,16},{13,19,22},{13,41,42}}
{{14,17,22},{14,31,34},{14,97,98}}
{{15,26,29},{15,55,56}}
{{16,23,28},{16,41,44},{16,127,128}}
{{17,14,21},{17,21,26},{17,34,37},{17,71,72}}
{{18,161,162}}
{{19,27,32},{19,43,46},{19,89,90}}
{{20,25,32},{20,65,68},{20,199,200}}
{{21,53,56},{21,109,110}}
{{22,31,38},{22,79,82},{22,241,242}}
{{23,29,36},{23,41,46},{23,64,67},{23,131,132}}
{{24,55,60},{24,287,288}}
{{25,35,42},{25,49,54},{25,76,79},{25,155,156}}
{{26,33,42},{26,65,70},{26,111,114},{26,337,338}}
{{27,89,92},{27,181,182}}
{{28,39,48},{28,129,132},{28,391,392}}
{29,37,46},{29,67,72},{29,103,106},{29,209,210}}
{{30,449,450}}
{{31,34,45},{31,43,52},{31,56,63},{31,77,82},{31,118,121},{31,239,240}}
{{32,41,52},{32,169,172},{32,511,512}}
{{33,64,71},{33,134,137},{33,271,272}}
{{34,47,58},{34,79,86},{34,113,118},{34,191,194},{34,577,578}}
{{35,45,56},{35,99,104},{35,151,154},{35,305,306}}
{{36,89,96},{36,127,132},{36,647,648}}
{{37,51,62},{37,111,116},{37,169,172},{37,341,342}}
{{38,49,62},{38,239,242},{38,721,722}}
{{39,71,80},{39,91,98},{39,188,191},{39,379,380}}
{{40,55,68},{40,265,268},{40,799,800}}
{{41,53,66},{41,64,75},{41,79,88},{41,101,108},{41,137,142},{41,208,211},{41,419,420}}
{{42,881,882}}
{{43,59,72},{43,71,82},{43,151,156},{43,229,232},{43,461,462}}
{{44,57,72},{44,103,112},{44,191,196},{44,321,324},{44,967,968}}
{{45,251,254},{45,505,506}}
{{46,63,78},{46,113,122},{46,209,214},{46,351,354},{46,1057,1058}}
{{47,61,76},{47,86,97},{47,134,141},{47,181,186},{47,274,277},{47,551,552}}
{{48,161,168},{48,1151,1152}}
{{49,50,69},{49,67,82},{49,94,105},{49,115,124},{49,146,153},{49,197,202},{49,298,301},{49,599,600}}
{{50,65,82},{50,175,182},{50,415,418},{50,1249,1250}}

It takes 765ms to calculate and print on screen all 9531 solutions with <=6 x <=1000
The 145 solutions above didn't register any time.

Paul.