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I believe the equations work as shown below. I just need to see the graph where y approaches zero. Also I can not use NSolve to find where the equations equal zero. As always it is my lack of good programming skills. So any advice is helpful.
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​​​​​​​​​​​​​​Clear[x,pnp];pnp=2564855351;​​((Sqrt[(41221^2*pnp^4+2*pnp*41221^5)+41221^8])/pnp^4-(1-(41221^2/(2*pnp)))*(pnp^2/41221^2))
-
8798838519304646211
3398341682
+
494652
300530820916134353551081538380157126
43276438207017172921441017973328906401
N-
8798838519304646211
3398341682
+
494652
300530820916134353551081538380157126
43276438207017172921441017973328906401

-2.58916×
9
10
​​​​Clear[x,pnp];pnp=2564855351;​​(Sqrt[(41221^2*pnp^4+2*pnp*41221^5)+41221^8])/pnp^4-(1-(41221^2/(2*pnp)))
-
3430539861
5129710702
+
494652
300530820916134353551081538380157126
43276438207017172921441017973328906401
N-
3430539861
5129710702
+
494652
300530820916134353551081538380157126
43276438207017172921441017973328906401

-0.668759
​​​​​​Clear[x,pnp];pnp=2564855351;​​x=41221;​​((Sqrt[(x^2*pnp^4+2*pnp*x^5)+x^8])/pnp^4-(1-(x^2/(2*pnp))))
N-
3430539861
5129710702
+
494652
300530820916134353551081538380157126
43276438207017172921441017973328906401

-0.668759
​​Clear[x,pnp];pnp=2152741102718889701896015201312825429257773588845675980170497676778133145218859135673011059773491059602497907111585214302079314665202840140619946994927570407753;​​​​x=4542789285848139407168619064973883165613714577846979325095998470925000415733​​5359;​​​​((Sqrt[(x^2*pnp^4+2*pnp*x^5)+x^8])/pnp^4-(1-(x^2/(2*pnp))))​​​​
4542789285848139407168619064973883165613714577846979325095998470925000415733
N-(4305482184800844908175381954762600651303060081927506027543419279375483131567190300127479540272286484367331032305590835761960487543956089656150267863632306888217/4305482205437779403792030402625650858515547177691351960340995353556266290437718271346022119546982119204995814223170428604158629330405680281239893989855140815506)+58√13175175544318586333287831791792295179865851611105469604666098412905564835985107926520946912772290035194332610056978066979392359992946856352967478527749595837675502805197819225809022332150298875120212715167155017034589949401075801959327767922586363725552534455819126345791138138405024619857888286511082866586407702399756969263529937663564149340176361422149722859594871949937414873457369718106176191762462748195919502610295670021381683011506391650230219313640146378769334801660953258193021825705227809501786176995687145552585230093767637809119334552405645073160023652108523059792001576418005589597906284691575393779219774300087727579830182593844101057882635811482668407815101707144949701637668149151975856351185866022711771471301144765499865098345047858915474440979511080392910561105519721476683245034144828471336371346039417063361453747054459444205771970111797151857233373132075790825703364339013539409911113953040050573107098908317919904169362482614698496417182870609333340197248100115336675385648171704728617160595881503683132860266407217538555329637187552222601126417458297909017854027928986419039226206575983045794484305627234679106960758391076707646171512806496341086184066468422509299243709027263801983717030601783156005205417279308234186617548958184132519845891428669640709138053166391097805161947513413377606577612965436750387113578871090942097705593279670152095073181424618985008515406273369847469716461735366162081
-1.
​​​​​​​​Clear[x,pnp];pnp=2519590847565789349402718324004839857142928212620403202777713783604366202070\​7595556264018525880784406918290641249515082189298559149176184502808489120072\​8449926873928072877767359714183472702618963750149718246911650776133798590957\​0009733045974880842840179742910064245869181719511874612151517265463228221686\​9987549182422433637259085141865462043576798423387184774447920739934236584823\​8242811981638150106748104516603773060562016196762561338441436038339044149526\​3443219011465754445417842402092461651572335077870774981712577246796292638635\​6373289912154831438167899885040445364023527381951378636564391212010397122822\​120720357;​​​​​​Plot[((Sqrt[(x^2*pnp^4+2*pnp*x^5)+x^8])/pnp^4-(1-(x^2/(2*pnp)))),{x,0,pnp}]
Plot
:Limiting value 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270240256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357 in {x,0,pnp} is not a machine-sized real number.
Plot
(
2
x
4
pnp
+2pnp
5
x
)+
8
x
4
pnp
-1-
2
x
2pnp
,{x,0,pnp}
​​​​​​​​​​​​​​Clear[x,pnp];pnp=2152741102718889701896015201312825429257773588845675980170497676778133145218859135673011059773491059602497907111585214302079314665202840140619946994927570407753;​​​​Plot[((Sqrt[(x^2*pnp^4+2*pnp*x^5)+x^8])/pnp^4-(1-(x^2/(2*pnp)))),{x,0,4542789285848139407168619064973883165613714577846979325095998470925000415733​​5359}]​​​​​​
5.0×
78
10
1.0×
79
10
1.5×
79
10
2.0×
79
10
0.2
0.4
0.6
0.8
1.0
Show[%64,ImageSizeFull]
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