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cmti10AprimePythagor}'eantrianglehasthreeintegersidesofwhichthehypotenuseand onele}'gareprimes.mInthisarticleweinvestigatetheirpropertiesanddistribution.Wear}'ealsointerestedinndingchainsofsuchtriangles,wherethehypotenuseofoneatriangleisthele}'gofthenextinthesequence.}WeexhibitachainofsevenprimePythagor}'eantrianglesandweincludeabriefdiscussionofprimalityproofsforthelar}'gerelements(upto2310digits)oftheassociatedsetofeightprimes.:9K`y
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cmcsc10Introduction WhileFinvestigatingthedistributionofspGecialformsofprimes,|therstauthorac-cidentlycameacrossaconjectureabGoutPythagoreantriangles(righttriangleswithintegralQsides).LTheconjecture,basedonthefamousConjecture(H)>ofSierpiGnskiandwSchinzel,>statesthatthereisaninnitenumbGerofPythagoreantriangleswhichhaveUUalegandhypGotenuseUUbothprime[html:color push cmyk 0 1 0 09 color pop html: ,page408]. Pythagorean?triangleshave?bGeenthesub 8jectofmuch?recreationalmaterial[html:color push cmyk 0 1 0 01 color pop html: ]aswell/asthebasisofsomeofthemostimpGortantandfundamentaltopicsinnumbGertheory*.4However,{we@couldnotndanysignicantreferencestosuchtwo-primePythagoreantriangles,)andhopingthatwehadfoundanewtopictostudyweenthusiasticallyUUstarted 6color push Black ٓR cmr71h color pop *7
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color push Black (1) color pop" developingUUappropriatetheoryandcomputerprograms; html: html: color push Black (2) color pop" searchingUUforlargetwo-primetriangles;html: html:color push Black (3) color pop" searchingforsequencesoftwo-primetriangleswherethehypGotenuseofthe " previousUUtrianglebGecomesthelegofthenextone.yP Thelargesttwo-primePythagoreantrianglethatwasfoundhadalegof5357digits?zandanhypGotenuse?zof10713digits.07Itsoonbecameapparentthatndingsequences^oftriangleswasexceptionallyinterestingandchallenging.LEventuallyasequence_ofseventriangleswasfound.Moresignicantthantheseventrianglesisthe]improvementbythesecondauthorofthegeneralmethoGd,APRCL,forprimalityprovingUUsothattheseventhhypGotenuseof2310digitscouldbGeprovedprime.1html: html:
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cmsy10 8v[ٟ2L;BG=2uv[;C~4=u2S+8v[ٟ2L;withxgcdx(u;v[ٲ)=1,Randxu;v_ofdierentparity*.YSinceA=(uPV+v[ٲ)(u v),RforxAto bGeUUprimeitisnecessarythat(u8 v[ٲ)=1UUsothaty>(A=2v+81;BG=2v[ٟ2,+2v[;C~4=2v2,+82v+1:ThusUUhtml: html:U(2.1) 5C~4=<$KA^2S+81Kw fe 5S (֍2"bѵ:pNotethattheevenlegisonlyonelessthanthehypGotenuse.IThetrianglesgetquitethinUUasAincreases. T*ondtwo-primePythagoreantrianglesitisnecessarytondpairsofprimesA;CHthat,satisfytheabGove,equation.GT*able1liststhesmallesttwo-prime,Pythago-rianUUtriangles. ;Ltcolor push Black ㌍S;=aT able1. html:color push gray 0 color pop html:PythagoreanUUtriangleswithtwoUUprimesides`qƍOEg ff ȕvfdͤ ffΟfdrank͡ ff# primeUUlegdhevenUUleg hypGotenuse͟ ff ff ȕvͤ ffџfd1͡ ffF_3 j4 u5͟ ffͤ ffџfd2͡ ffF_5}i12 t13͟ ffͤ ffџfd3͡ ffA^11}i60 t61͟ ffͤ ffџfd4͡ ffA^19xh180 s181͟ ffͤ ffџfd5͡ ffA^29xh420 s421͟ ffͤ ffџfd6͡ ffA^59sg1740 r1741͟ ffͤ ffџfd7͡ ffA^61sg1860 r1861͟ ffͤ ffџfd8͡ ffA^71sg2520 r2521͟ ffͤ ffџfd9͡ ffA^79sg3120 r3121͟ ffͤ ffПfd10͡ ff<]101sg5100 r5101͟ ff ff ȕvͤ ff ϟfd100͡ ff7\4289dd9197760 o9197761͟ ff ff ȕvͤ ffΟfd1000͡ ff2[91621Ua4197203820 l4197203821͟ ff ff ȕv color pop! SmallNtrianglesareeasytondbyasimplesearch,'Lbutndinglargetriangleswiththousandsofdigitsiscomplicatedbythedicultyofprovingtrueprimality 6color push Blackh color pop 67
6color push Blackhtml:color push gray 0 color pop html:d3h color popV 6लof`thehypGotenuse,+C .WvHowever,if`(CR 1)hasmanyfactorsthenitiseasytoprove@6primality-using[html:color push cmyk 0 1 0 02 color pop html: ],5assumingthatthefactoredpartof(C ߲1)exceedsZ cmr53ܟp#3 fe ̟wvC.dSincehtml: html:w6(2.2)sƵC 81=<$KA^2S+1Kw fe 5S (֍2$ 1=(A2S 1)=2=(A 1)(A+1)=2;ˍ6लbypickinganappropriateformforA,then(AZZ 1)canbGecompletelyfactoredso 6thatUU(C 81)willbGeabout50%factored.BUsingtheformA=k 10^ 0er cmmi7n'+1,acomputersearchofafewdaysgavethefollowing6largeUUtriangle:Xj$A=4911408101300+1;1306UUdigits, C~4=2612digits.BAUfewydaysafterthisresultwaspGostedtotheNMBR*THRYUlistywereceiveda 6messageUUfromIagoCambGoaannouncingamuchlargertriangle:k{ A=14918217783+1;5357UUdigits,UWC~4=10713digits.6HeBcleverlyusedapreviouslycomputedlistofprimesasasourceforAthuselimi-6natingUUthelargeamountoftimerequiredtondtherstprime.6ट
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9;U3.H_sTwo-primePythagoreantrianglesequences ItUUispGossibletondaseriesofprimes,P0|s;P1;P2;:::;Pk됵;:::;PnӲsuchUUthathtml: html:2>(3.1) @ϵPk+B+1ҫ=<$KP^c2vk+81Kw fe o (֍B72":&This^prepresentsasequenceofntwo-primetriangleswherePkJ isthehypGotenuseofthekP-thtriangleandthelegofthe(k?+1)-thtriangle.mLEachPbhasabGouttwicetheDtnumbGerofdigitsasthepreviousPc.l'T*able2isalistofthesmallestsetsoftwosequentialUUprimePythagoreantriangles. 獠|@color push Black ;T able2. html:color push gray 0 color pop html:TwoUUsequentialprimePythagoreantrianglesH`"L͉ ff" 3fd͟ ff ff.%triangleUU1s ff '=triangleUU2? ff ff" 3ͤ ffΟfd1͡ ff 3F 4n 5͟ ff 5 %12 113͟ ffͤ ffΟfd2͡ ff 11A 60i 61͟ ff 61 #1860 /1861͟ ffͤ ffΟfd3͡ ff 19<