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Prime Fibonacci numbers (Posted on 2013-05-03) Difficulty: 3 of 5
3 and 5 are twin primes, and the 3rd and 5th Fibonacci numbers are both prime. 5 and 7 are twin primes, and the 5th and 7th Fibonacci numbers are both prime. 11 and 13 are twin primes, and the 11th and 13th Fibonacci numbers are both prime. Are there any other twin prime pairs (p, p+2) such that the pth and (p+2)th Fibonacci numbers are both prime?

See The Solution Submitted by Math Man    
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Some Thoughts Exploration | Comment 1 of 3

 10   OldFib=1:Fib=1:NewFib=2:FibNo=2
 20   while Ct<47
 30     NxtFib=Fib+NewFib
 40     OldFib=Fib:Fib=NewFib:NewFib=NxtFib
 50     inc FibNo
 60     if prmdiv(OldFib)=OldFib and prmdiv(NewFib)=NewFib then
 70        :print FibNo-1;OldFib,FibNo+1;NewFib,:inc Ct
 80        :if prmdiv(FibNo-1)=FibNo-1 and prmdiv(FibNo+1)=FibNo+1 then
 90          :print "*":else print
100   wend

looks for prime Fibonacci numbers with indices differing by 2. If those indices are also prime, the pair are marked with an asterisk:

 2  1             4  3
 3  2             5  5   *
 5  5             7  13  *
 11  89          13  233 *
 


The first one, a non-solution, is also spurious in the fact that the Fibonacci number 1 is not actually prime, but the "largest prime factor" function returns a 1 for the argument 1, making it seem prime.

These are the only results found up to where the interpreter reports

Illegal parameter in 60

as the next Fibonacci number it was to check for primality via the prmdiv function was Fib(12473) =

 2256066183207603328903017542806442384760157830296163134812558868863970689154873
38569216615294378807772264848648240663960114406505001695290636829787920778575396
41846358528147669294932246732383135510025381869525148533744809974794737930386347
84418342137255479113299498483735260116083096045835662745964569490620018826158568
33993538371663456530932920463736092158797133797659734052421058454963257202221720
68022472433217971862864592721761657868204850535819109843399832325695033254447263
20812886211316106005982490952164214906146006835048139320435446770686902770280990
80155594533980947518108718602889771461040134688769059832578935782891251004377261
94225020594511499144977486477003897323347259210567551366163649941054042945525662
27703240037584838473466045654653862214783054875957814625589177522691696753913752
54161394110207767612816165547934814597459795242491200494129284589334901866458235
03342691244403790484413595029505764985589512385599510591570001757230963408404452
94735746548542670170213207731054298516945404805629527759882090193663532433746056
75626469124627505519441703604282993885732918787652698593466657653701072191217076
66101640977489240091615019317906052461594762750828452845050712435089055319953796
36248252973253169350346846282517969883869605876977483778902188802950288436215175
50387441273285720339291626378814734831304389369747767870958973400765722589656235
06434078570387763207185918266554232540310798736716779105646609866488948890505797
53667474886546318499926372486900622184993176355789621395666999032136479160959823
82945756791349191044427344777462922568314460684581569604980416802008622207798903
96439191839709198189324637370449945273676864795039713975375204722489552337412763
32033507342199178941264516887742432959729772672773769306512234113291191953996995
94073302355778213505426915231348083215468049616580535253996770453352470249448483
07744597144658886339178198892388240374334438363105829194405690622874636103507435
34996496983325949279827212786210850734386745485812278892522241397537277297563566
67580225311002302953857710459732279871433272831050172883157400750870259297629781
21984227091852679700807144789591137823585522493966410843365443546248242891136468
97693008850658838476744526875180570698915460940014452203823097085615058000739238
19443343679832962545710712767741864841243295238786218532196436922568033159149954
57761782355055087470353355369576519736781766551651264064624839703561552320131169
11798214293121716064952163392076961892404007397363159356083138740908169533534987
84652963772632014848585036905641638480461968724705089430865069911595895777338462
522605022278778924228376931995457726743066193293

exceeding UBASIC's limit on its argument to the prmdiv function.


 


  Posted by Charlie on 2013-05-03 17:09:33
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