1.Introduction

The problem of primitive Pythagorean triangles with same perimeter is mentioned in Guy's book[1],D21.
According to Guy's book, Beiler gave four primitive Pythagorean triangles with perimeter 317460.

We show the smallest solutions for primitive Pythagorean triangles such that {5,6,7,8,9,10} triangles have same perimeters.
Furthermore, we show the sets of 34 primitive Pythagorean triangles.(may be smallest)

2. Results

A^2 + B^2 = C^2
A=(x^2-y^2), B=2xy, C=(x^2+y^2), perimeter = A+B+C = 2x(x+y).

   A       B       C     perimeter
[81003,   728596,   733085,  1542684]
[263523,  612436,   666725,  1542684]
[496155,  405652,   640877,  1542684]
[626835,  243412,   672437,  1542684]
[734483,   70356,   737845,  1542684]

[752505,  5468848, 5520377, 11741730]
[924209,  5369280, 5448241, 11741730]
[2723721, 4097680, 4920329, 11741730]
[3873705, 2980432, 4887593, 11741730]
[4250561, 2539680, 4951489, 11741730]
[5831969,   77280, 5832481, 11741730]

[1136499, 19696540, 19729301, 40562340]
[6164115, 16646812, 17751413, 40562340]
[8288579, 15072540, 17201221, 40562340]
[13657371, 9986140, 16918829, 40562340]
[14725451, 8722140, 17114749, 40562340]
[17088315, 5517148, 17956877, 40562340]
[19023459, 2368540, 19170341, 40562340]

[1905225,  49732072, 49768553, 101405850]
[5855225,  47595912, 47954713, 101405850]
[17188521, 40354600, 43862729, 101405850]
[21750225, 36858328, 42797297, 101405850]
[33985329, 25144600, 42275921, 101405850]
[36200225, 22554168, 42651457, 101405850]
[45530225,  9387672, 46487953, 101405850]
[46215809,  8244600, 46945441, 101405850]

[3515379,  117800380, 117852821, 239168580]
[24926811, 105670780, 108570989, 239168580]
[46264259,  90904380, 101999941, 239168580]
[63791531,  76086780,  99290269, 239168580]
[86868299,  51376380, 100923901, 239168580]
[88983555,  48731452, 101453573, 239168580]
[102629355, 29699068, 106840157, 239168580]
[111365555, 15380412, 112422613, 239168580]
[118937355,  1286908, 118944317, 239168580]

[220735515,   4665412, 220784813, 446185740]
[60652515,  187995652, 197537573, 446185740]
[213653739,  18111940, 214420061, 446185740]
[155721891, 103489540, 186974309, 446185740]
[212248715,  20682948, 213254077, 446185740]
[175747715,  78113028, 192324997, 446185740]
[130590515, 130779012, 184816213, 446185740]
[5395715,   220361988, 220428037, 446185740]
[156607451, 102441540, 187136749, 446185740]
[84463379,  170999940, 190722421, 446185740]


[81297321105, 31922885872, 87340283153, 200560490130]
[10956850905, 94485231328, 95118407897, 200560490130]
[61382995905, 56052534928, 83124959297, 200560490130]
[6172089105,  97096214512, 97292186513, 200560490130]
[97556080881, 5304236080,  97700173169, 200560490130]
[71894082729, 44247311680, 84419095721, 200560490130]
[12016087041, 93889304080, 94655099009, 200560490130]
[51818369889, 65344889680, 83397230561, 200560490130]
[91377166089, 16354199680, 92829124361, 200560490130]
[60544579161, 56917850080, 83098060889, 200560490130]
[40566090081, 74854496080, 85139903969, 200560490130]
[25661722449, 85566797680, 89331970001, 200560490130]
[705893529,   99926051680, 99928544921, 200560490130]
[67106902161, 49854500080, 83599087889, 200560490130]
[48064773081, 68673146080, 83822570969, 200560490130]
[5001077289,  97715759680, 97843653161, 200560490130]
[80764896905, 32672385408, 87123207817, 200560490130]
[95315397905, 9461269488,  95783822737, 200560490130]
[33242678105, 80356580832, 86961231193, 200560490130]
[93078146105, 13439011872, 94043332153, 200560490130]
[77741783105, 36804857232, 86013849793, 200560490130]
[43058661905, 72865114608, 84636713617, 200560490130]
[20585154905, 88810426848, 91164908377, 200560490130]
[3819452105,  98333444352, 98407593673, 200560490130]
[76307765105, 38694783792, 85557941233, 200560490130]
[58578482105, 58906942752, 83075065273, 200560490130]
[45352068905, 70978248768, 84230172457, 200560490130]
[23205801905, 87159173808, 90195514417, 200560490130]
[33610616105, 80091665472, 86858208553, 200560490130]
[96022727801, 8168242080,  96369520249, 200560490130]
[47373575249, 69268237680, 83918677201, 200560490130]
[9209361161,  95453950080, 95897178889, 200560490130]
[77001913049, 37785427680, 85773149401, 200560490130]
[46219153361, 70250260080, 84091076689, 200560490130]

3.Reference

[1] R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer-Verlag, 1994.