Determine all possible prime number(s) N such that (7^N-1 – 1)/N is a positive perfect square.

Prove that these are the only value(s) of N that satisfy the given conditions.

The only prime under 3049 that works is 3. The following program checks up to that value, when the value of (7^(N-1)-1)/N becomes a 2573-digit number, preventing UBASIC from taking its square root via built-in function.

list
    5   N=1
   10   while N>0
   20     N=nxtprm(N)
   30     V=(7^(N-1)-1)//N
   40     if V=int(V) then
   50        :Sr=int(sqrt(V)+0.5)
   60        :if Sr*Sr=V then print N;V;Sr
   70   wend
OK
run
 3  16  4
Overflow in 50
?n,v
 3049    23695645235380467868868309285488693124310463191804539500437171340734782
74809622359177041626597303927825927993229036294093875689144012519091331336658741
18091008093987994099881596827945875004821563597955805406258730706639103070568558
99995024113449304679655735987985420907852768321429628485903406058488842208955767
74231603477334780079775513550512561580948240317111475984223106457207788376165168
94711006300805225713888009050723115722279852543997553978380968799728120091813057
76643490295506732856571386249648360680072483583587029283396053844872106083467116
36288101285776089550244220462859019170185977426548386787112086744401973117372239
84055526360037784994999318701527504013133080544611852206210540172577799409264708
48968034290001380993744265710162584674144841700891858545722327384299273526734543
08378969907884240109386679704112616494361541528993826889110912045895342204845832
73322841065836284068533367136213731226918111837521875379666753400731481053630099
34966132565094933473574742805783016045702459997258641642471734070164136737486860
33280055585813542661545024994912673523371109691905025814324112217107978944167952
54953780380422223554457391447069130000777704707246254754616471419989909254757071
07285402262763454934713802961031468762887546763130515770967804076810001433192934
35772343604234156787608530862884195640522923231224240807587471476738255508801381
46135850557954794284316224755511974529301678308995166677323621496053524314279282
78927454899408107081566627217075485133981729223704622081180446238137628932340358
47941818696052923291286004658217843056776645975147275100518417859746721704298898
66690997363948230701888119627119668768105647678736710694715345185074659056659494
28090489233028236733040453074313546378659213308452417161517215338326297092625872
66831472841821005005613446565794220753397073270033383735576803802707040374086143
72285946290413786265085380740465186376382472938064071466741983645271273665910465
56189294829319989357029462824789420057963766947272303385892804459978821730671419
39889196503391809825168697743959322342866293968515509292393723813585486636222285
90108906172667581857998648931625481341228656868806479145307134032729993267891711
68733166193338391759197160741747798605501343548346605909714994158464842241189128
06336613046720877282842109769013071716194693686876658467990567309907571880186351
42614829309747731487865774253886296715330168598288017837720713554739972334019512
40657767146248282621194914625997802990821948697492046445753119771111246057032699
20250584974764291024667040842060198150372459296350056581634117996929966039631883
0706207847410180171200
OK

Posted by Charlie on 2010-04-24 13:43:39