Keywords: Heptic non-homogeneous equation, Pyramidal numbers, Pronic numbers, fourth dimensional figurate numbers.

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1 [Gol 5: M 0] ISSN: IJEST INTENTIONL JOUNL OF ENGINEEING SCIENCES & ESECH TECHNOLOGY O the Hetc No-Hoogeeous Euto th Four Ukos z 6 0 M..Gol * G.Suth S.Vdhlksh * Dertet of MthetcsShrt Idr Gdh CollegeTrch Id lgol@gl.co strct The hetc o-hoogeeous euto th four ukos rereseted the dohte euto z 6 0 s lzed for ts tters of o-zero dstct tegrl solutos d to dfferet ethods of tegrl solutos re llustrted. Vrous terestg reltos etee the solutos d secl uers el Prdl uers Proc uers Str uers Gooc uers Cetered Polgol uers Cetered Hegol Prdl uers fourth desol fgurte uers re ehted. Keords: Hetc o-hoogeeous euto Prdl uers Proc uers fourth desol fgurte uers. MSC 000 Mthetcs suject clssfcto: D. NOTTIONS: T : Polgol uer of rk th sze S : Str uer of rk P : Proc uer of rk Goc : Gooc uer of rk Ct : Cetered olgol uer of rk th sze. Cf 6 : Cetered Heogol Prdl uer of rk Cf : Fourth desol trgulr fgurte uer of rk Itroducto The theor of dohte eutos offers rch vret of fsctg roles. I rtculr hoogeeous d o-hoogeeous eutos of hgher degree hve roused the terest of uerous Mthetcs sce tut [-].Prtculrl [5] secl eutos of sth degree th four d fve ukos re studed. I [6-8] hetc eutos th three d fve ukos re lsed.ths coucto lses hetc euto th four ukos gve z 6 0. Iftel o-zero teger udtules z stsfg the ove euto re oted. Vrous terestg roertes og the vlues of z d re reseted. Method of lss The dohte euto reresetg o-hoogeeous hetc euto s z 6 0 To strt tht s oserved tht s stsfed the follog teger udrules z : htt: //.jesrt.co C Itertol Jourl of Egeerg Sceces & eserch Techolog [-7]

2 [Gol 5: M 0] ISSN: Itroducg the ler trsfortos u v u v z u t leds to 6 v u c e rtte s v u Ptter: Note tht s slr to the ell ko Pthogore euto.usg the ost cted soluto of stdrd Pthogore eutoe hve Note tht 6 s stsfed u... v here > here > 0 7 d thus fro 5e hve u... v Susttutg 8 the corresodg vlues of z re gve... z Thus 9 d 7 reresets the o-zero dstct teger soluto to. Proertes:.Ech of the follog s st uer. [ ]. 8[ z ] c. [ z ]. z Cf 6Cf S 8t Ct t Cf 6 Pr Goc t t 0od s erfect sure. 5. Ech of the follog s cuc teger t Pr t. Goc t t 6. z s udrtc teger htt: //.jesrt.co C Itertol Jourl of Egeerg Sceces & eserch Techolog [-7]

3 [Gol 5: M 0] ISSN: htt: //.jesrt.co C Itertol Jourl of Egeerg Sceces & eserch Techolog [-7] Ptter: 6 s solved through dfferet rocess s sho elo: Tke here d elog the ethod of fctorzto t s rtte s the eresso of doule eutos hch s stsfed Fro 5 d e get... z Thus 5 0 d rereset the o-zero tegrl solutos to. Ptter: s rertte s v u ssue v v 0 d rte s 0 Susttutg d elog the ethod of fctorztodefe u 5 fro hch e get u 6 7 here 8 choose d such tht 9 d thus 0 Tretg 9 s udrtc d solvg for e hve ± Ths sure root o the rght hd sde of the ove euto s reoved relcg

4 [Gol 5: M 0] ISSN: htt: //.jesrt.co C Itertol Jourl of Egeerg Sceces & eserch Techolog [-7] g relcg resectvel... the ove euto t ecoes hch s stsfed 5 6 g tkg 7 e hve 8 9 Fro 8 d 7 e hve Susttutg the ove vlues of d d tkg the ostve vlue of the sure root e hve Susttutg the vlues of d 8e get Usg the vlues of d 06 d the corresodg tegrl solutos re rereseted

5 [Gol 5: M 0] ISSN: z cosderg the egtve vlue for the sure root o the rght hd sde of other soluto tter s oted. For eletkg d follog the lss reseted ove the corresodg to sets of ozero tegrl solutos to re gve Set 5 00d d z 5 50d 00d 500d Set 5 5 z 5 50d 0000d 0d 0000d 5d Cocluso To coclude oe serch for other tter of solutos d ther corresodg roertes. efereces [] L.E.Dckso Hstor of Theor of Nuers Vol. Chelse Pulshg cone York 95. [] L.J.Mordell Dohte eutos cdec Press Lodo969 [] Crchel.D.The theor of uers d Dohte lssdover Pulctos Ne York 959 [] M..GolS.Vdhlksh d K.Lksh O the o-hoogeeous setc euto z IJM7-70 [5] M..GolS.Vdhlksh d K.Lksh Itegrl Solutos of the setc euto th fve 5 ukos z T IJEST50-50 Dec.0 [6] M..Gol d sgeeth.g retrc tegrl solutos of the hetc euto th 5ukos 5 X Y z essel Jourl of Mthetcs 7-0. [7] M..Gol d sgeeth.g O the hetc dohte eutos th 5 5 ukos X Y z [8] Mjusoth G.sgeeth d M..Gol O the o-hoogeeous 7 5 hetc eutos th ukos z of Mthetcs 7-0 trctc Jourl of Mthetcs Dohte jourl htt: //.jesrt.co C Itertol Jourl of Egeerg Sceces & eserch Techolog [-7]

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