**(1)**

Find a Pythagorean triangle whose area contains all ten digits and one of whose sides contains only a single repeated digit.

**(2)**

Find a Pythagorean triangle whose area contains only a single repeated digit and one of whose sides contains nine or ten distinct digits.

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## About benvitalis

math grad - Interest: Number theory

For part 1 there are these, format {a, b, c, area}, part 2 isn’t looking too good atm.

{111111,3816648,3818265,212035787964}

{111111,17004852,17005215,944713055286}

{111111,62351740,62351839,3463982091570}

{222222,1212120,1232322,134679865320}

{222222,28777320,28778178,3197476802520}

{222222,53444160,53444622,5938234061760}

{222222,111222000,111222222,12357987642000}

{333333,230880,405483,38479961520}

{333333,1110444,1159395,185073814926}

{333333,1918056,1946805,319675680324}

{333333,2153844,2179485,358973641026}

{333333,3917160,3931317,652859347140}

{333333,5494944,5505045,915823084176}

{333333,9958740,9964317,1659788340210}

{333333,43165980,43167267,7194322805670}

{333333,721499240,721499317,120249753083460}

{333333,1133784556,1133784605,188963903702574}

{444444,456192,636900,101375898624}

{444444,583083,733155,129573870426}

{444444,622935,765231,138429861570}

{444444,743625,866319,165249834750}

{444444,759583,880055,168796053426}

{444444,865467,972915,192325807674}

{444444,1132560,1216644,251679748320}

{444444,1461915,1527981,324869675130}

{444444,2424240,2464644,538719461280}

{444444,3663333,3690195,814073185926}

{444444,4416192,4438500,981375018624}

{444444,7628445,7641381,1695208304790}

{444444,9073917,9084795,2016423983574}

{444444,11025815,11034769,2450178660930}

{444444,8230436208,8230436220,1828983995014176}

{555555,444600,711555,123499876500}

{555555,484848,737373,134679865320}

{555555,551772,783003,153269846730}

{555555,883376,1043549,245381976840}

{555555,1180260,1304475,327849672150}

{555555,1689740,1778725,469371752850}

{555555,6350300,6374555,1763970458250}

{555555,7504740,7525275,2084647915350}

{555555,9714276,9730149,2698407301590}

{555555,18051264,18059811,5014234985760}

{555555,44535260,44538725,12370893184650}

{555555,83011808,83013667,23058812496720}

{555555,2374164260,2374164325,659489412732150}

{666666,1128688,1310870,376228957104}

{666666,1416312,1565370,472103527896}

{666666,1749840,1872534,583279416720}

{666666,2027688,2134470,675895324104}

{666666,73049688,73052730,24349871650104}

{666666,37036962960,37036962966,12345641974345680}

{777777,387464,868945,150680293764}

{777777,1362636,1568985,529913470086}

{777777,2367036,2491545,920513079486}

{777777,4242420,4313127,1649828350170}

{777777,5641664,5695025,2193978250464}

{777777,23237060,23250073,9036625407810}

{777777,436462180,436462873,169735122486930}

{777777,474832540,474833177,184656914231790}

{777777,1789754536,1789754705,696014956873236}

{777777,14403263364,14403263385,5601263484730914}

{777777,43209790120,43209790127,16803790465081620}

{888888,30555,889413,13579986420}

{888888,35334,889590,15703984296}

{888888,154791,902265,68795931204}

{888888,308385,940863,137059862940}

{888888,415584,981240,184703815296}

{888888,530816,1035320,235917986304}

{888888,559584,1050360,248703751296}

{888888,1060605,1383837,471379528620}

{888888,1245870,1530462,553719446280}

{888888,1888416,2087160,839295160704}

{888888,2095830,2276538,931479068520}

{888888,2803009,2940575,1245780531996}

{888888,3646305,3753087,1620578379420}

{888888,3958720,4057288,1759429351680}

{888888,7926309,7975995,3522800477196}

{888888,9375795,9417837,4167015832980}

{888888,11789525,11822987,5239783649100}

{888888,13267166,13296910,5896512325704}

{888888,15060955,15087163,6693751084020}

{888888,18656066,18677230,8291576597304}

{888888,19424160,19444488,8632951367040}

{888888,30326791,30339815,13478560299204}

{888888,31771530,31783962,14120665879320}

{888888,35413191,35424345,15739180260804}

{888888,48696750,48704862,21642978357000}

{888888,53440695,53448087,23751396248580}

{888888,71256416,71261960,31669486552704}

{888888,111220335,111223887,49431210568740}

{888888,148294816,148297480,65908741202304}

{888888,249406366,249407950,110847162930504}

{888888,292204160,292205512,129868385687040}

{888888,310094291,310095565,137819547069204}

{888888,333665741,333666925,148295736593004}

{888888,345332416,345333560,153480920296704}

{888888,723554559,723555105,321579482420196}

{888888,2170664405,2170664587,964738770815820}

{888888,4115218059,4115218155,1828983975014196}

{99999,31360,104801,1567984320}

{99999,1503880,1507201,75193248060}

Paul.

I’ve also found the same results for question#1 by running a program. Some of these puppies don’t have m,n generators !? Do you have an algorithm to find them?

I decided to throw question#2 in, not sure whether there’s a solution

I use my algorithm of finding pythagoean triangles, I just increment the repdigit and find all the PT’s with said small side, I then just calculate the area and test if it contains all the digits. As soon as it find a solution it increments the digit. The above results took 31ms to find.

Regarding #2 I tested all repdigits upto 50 digits in length for all digits 1 – 9 and the only solution with any PT was with the 6 digit with {693, 1924, 2045, 666666}.

Paul