# first primitive Pythagorean triplets

$(\mbox{odd cathetus})^{2}+(\mbox{even cathetus})^{2}=(\mbox{hypotenuse})^{2}$

$3^{2}+4^{2}=5^{2}$ ($\leftarrow$ these form the so-called Egyptian triangle, known by the pyramid builders)
$5^{2}+12^{2}=13^{2}$
$15^{2}+8^{2}=17^{2}$
$7^{2}+24^{2}=25^{2}$
$21^{2}+20^{2}=29^{2}$
$9^{2}+40^{2}=41^{2}$
$35^{2}+12^{2}=37^{2}$
$11^{2}+60^{2}=61^{2}$
$45^{2}+28^{2}=53^{2}$
$33^{2}+56^{2}=65^{2}$
$13^{2}+84^{2}=85^{2}$
$63^{2}+16^{2}=65^{2}$
$55^{2}+48^{2}=73^{2}$
$39^{2}+80^{2}=89^{2}$
$15^{2}+112^{2}=113^{2}$
$77^{2}+36^{2}=85^{2}$
$65^{2}+72^{2}=97^{2}$
$17^{2}+144^{2}=145^{2}$
$99^{2}+20^{2}=101^{2}$
$91^{2}+60^{2}=109^{2}$
$51^{2}+140^{2}=149^{2}$
$19^{2}+180^{2}=181^{2}$
$117^{2}+44^{2}=125^{2}$
$105^{2}+88^{2}=137^{2}$
$85^{2}+132^{2}=157^{2}$
$57^{2}+176^{2}=185^{2}$
$21^{2}+220^{2}=221^{2}$
$143^{2}+24^{2}=145^{2}$
$119^{2}+120^{2}=169^{2}$
$95^{2}+168^{2}=193^{2}$
$23^{2}+264^{2}=265^{2}$
$165^{2}+52^{2}=173^{2}$
$153^{2}+104^{2}=185^{2}$
$133^{2}+156^{2}=205^{2}$
$105^{2}+208^{2}=233^{2}$
$69^{2}+260^{2}=269^{2}$
$25^{2}+312^{2}=313^{2}$
$195^{2}+28^{2}=197^{2}$
$187^{2}+84^{2}=205^{2}$
$171^{2}+140^{2}=221^{2}$
$115^{2}+252^{2}=277^{2}$
$75^{2}+308^{2}=317^{2}$
$27^{2}+364^{2}=365^{2}$
$221^{2}+60^{2}=229^{2}$
$209^{2}+120^{2}=241^{2}$
$161^{2}+240^{2}=289^{2}$
$29^{2}+420^{2}=421^{2}$
$255^{2}+32^{2}=257^{2}$
$247^{2}+96^{2}=265^{2}$
$231^{2}+160^{2}=281^{2}$
$207^{2}+224^{2}=305^{2}$
$175^{2}+288^{2}=337^{2}$
$135^{2}+352^{2}=377^{2}$
$87^{2}+416^{2}=425^{2}$
$31^{2}+480^{2}=481^{2}$
$285^{2}+68^{2}=293^{2}$
$273^{2}+136^{2}=305^{2}$
$253^{2}+204^{2}=325^{2}$
$225^{2}+272^{2}=353^{2}$
$189^{2}+340^{2}=389^{2}$
$145^{2}+408^{2}=433^{2}$
$93^{2}+476^{2}=485^{2}$
$33^{2}+544^{2}=545^{2}$
$323^{2}+36^{2}=325^{2}$
$299^{2}+180^{2}=349^{2}$
$275^{2}+252^{2}=373^{2}$
$203^{2}+396^{2}=445^{2}$
$155^{2}+468^{2}=493^{2}$
$35^{2}+612^{2}=613^{2}$
$357^{2}+76^{2}=365^{2}$
$345^{2}+152^{2}=377^{2}$
$325^{2}+228^{2}=397^{2}$
$297^{2}+304^{2}=425^{2}$
$261^{2}+380^{2}=461^{2}$
$217^{2}+456^{2}=505^{2}$
$165^{2}+532^{2}=557^{2}$
$105^{2}+608^{2}=617^{2}$
$37^{2}+684^{2}=685^{2}$
$399^{2}+40^{2}=401^{2}$
$391^{2}+120^{2}=409^{2}$
$351^{2}+280^{2}=449^{2}$
$319^{2}+360^{2}=481^{2}$
$279^{2}+440^{2}=521^{2}$
$231^{2}+520^{2}=569^{2}$
$111^{2}+680^{2}=689^{2}$
$39^{2}+760^{2}=761^{2}$
$437^{2}+84^{2}=445^{2}$
$425^{2}+168^{2}=457^{2}$
$377^{2}+336^{2}=505^{2}$
$341^{2}+420^{2}=541^{2}$
$185^{2}+672^{2}=697^{2}$
$41^{2}+840^{2}=841^{2}$
$483^{2}+44^{2}=485^{2}$
$475^{2}+132^{2}=493^{2}$
$459^{2}+220^{2}=509^{2}$
$435^{2}+308^{2}=533^{2}$
$403^{2}+396^{2}=565^{2}$
$315^{2}+572^{2}=653^{2}$
$259^{2}+660^{2}=709^{2}$
$195^{2}+748^{2}=773^{2}$
$123^{2}+836^{2}=845^{2}$
$43^{2}+924^{2}=925^{2}$
$525^{2}+92^{2}=533^{2}$
$513^{2}+184^{2}=545^{2}$
$493^{2}+276^{2}=565^{2}$
$465^{2}+368^{2}=593^{2}$
$429^{2}+460^{2}=629^{2}$
$385^{2}+552^{2}=673^{2}$
$333^{2}+644^{2}=725^{2}$
$273^{2}+736^{2}=785^{2}$
$205^{2}+828^{2}=853^{2}$
$129^{2}+920^{2}=929^{2}$
$45^{2}+1012^{2}=1013^{2}$
$575^{2}+48^{2}=577^{2}$
$551^{2}+240^{2}=601^{2}$
$527^{2}+336^{2}=625^{2}$
$455^{2}+528^{2}=697^{2}$
$407^{2}+624^{2}=745^{2}$
$287^{2}+816^{2}=865^{2}$
$215^{2}+912^{2}=937^{2}$
$47^{2}+1104^{2}=1105^{2}$
$621^{2}+100^{2}=629^{2}$
$609^{2}+200^{2}=641^{2}$
$589^{2}+300^{2}=661^{2}$
$561^{2}+400^{2}=689^{2}$
$481^{2}+600^{2}=769^{2}$
$429^{2}+700^{2}=821^{2}$
$369^{2}+800^{2}=881^{2}$
$301^{2}+900^{2}=949^{2}$
$141^{2}+1100^{2}=1109^{2}$
$49^{2}+1200^{2}=1201^{2}$
$675^{2}+52^{2}=677^{2}$
$667^{2}+156^{2}=685^{2}$
$651^{2}+260^{2}=701^{2}$
$627^{2}+364^{2}=725^{2}$
$595^{2}+468^{2}=757^{2}$
$555^{2}+572^{2}=797^{2}$
$451^{2}+780^{2}=901^{2}$
$387^{2}+884^{2}=965^{2}$
$315^{2}+988^{2}=1037^{2}$
$235^{2}+1092^{2}=1117^{2}$
$147^{2}+1196^{2}=1205^{2}$
$51^{2}+1300^{2}=1301^{2}$
$725^{2}+108^{2}=733^{2}$
$713^{2}+216^{2}=745^{2}$
$665^{2}+432^{2}=793^{2}$
$629^{2}+540^{2}=829^{2}$
$533^{2}+756^{2}=925^{2}$
$473^{2}+864^{2}=985^{2}$
$329^{2}+1080^{2}=1129^{2}$
$245^{2}+1188^{2}=1213^{2}$
$53^{2}+1404^{2}=1405^{2}$

N.B. that the lengths of the even cathetus and the hypotenuse are consecutive integers (as 1404 and 1405) always when the corresponding seed numbers $m$ and $n$ (see the parent (http://planetmath.org/PythagoreanTriplet) entry) are successive integers.

 Title first primitive Pythagorean triplets Canonical name FirstPrimitivePythagoreanTriplets Date of creation 2013-10-30 13:12:58 Last modified on 2013-10-30 13:12:58 Owner pahio (2872) Last modified by pahio (2872) Numerical id 12 Author pahio (2872) Entry type Example Classification msc 01A16 Classification msc 11-00 Synonym least coprime Pythagorean triplets Related topic PythagorasTheorem Related topic IncircleRadiusDeterminedByPythagoreanTriple Related topic ContraharmonicMeansAndPythagoreanHypotenuses Related topic PythagoreanHypotenusesAsContraharmonicMeans Defines Egyptian triangle