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May 9th, 2013, 11:06 PM   #1
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Conjecture on cycle length and primes : prime abc conjecture



Thanks for Charles at math.stackexchange convert my language to readable math language .

Suppose a>9 is odd and b is the cycle length of a as defined below. Then I conjecture that if

[attachment=1:2rcl19rx]c.jpg[/attachment:2rcl19rx]

for some positive integer c then a is prime.

Cycle length:

a is an odd number > 9 and Od is the odd part function, Sloane's A000265.

Let a(0) = 1 and a(n) = Od(a(n-1) + a) for n > 0. If a(n) = 1 for 0 < n < N/2 - 1 then the cycle length is the smallest such n, otherwise it is a/2 - 1.

For more information see A179382.

Example:

11 = 5*2^1+1

11 (1,3, 7, 9, 5)

97 = 24*2^2+1

97 (1,49, 73, 85, 91, 47, 9, 53, 75, 43, 35, 33, 65, 81, 89, 93, 95, 3, 25, 61, 79, 11, 27, 31)

if c = 1 that's OEIS A001122

if c = 2 that's OEIS A155072

if c = 3 that's OEIS A001134

if c = 4 that's 1217, 1249, 1553, 4049, 4273, 4481, 4993..., not in the OEIS

PARI/GP code:

[attachment=0:2rcl19rx]Pari_GP.jpg[/attachment:2rcl19rx]
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File Type: jpg Pari_GP.jpg (25.2 KB, 365 views)
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May 10th, 2013, 04:43 AM   #2
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Re: Conjecture on cycle length and primes : prime abc conjec

That was me. I did my best to explain it and I voted to reopen the question after it had been closed initially.

But I still don't have much insight into what this cycle length is and without that I'm stumbling around in the dark.
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May 10th, 2013, 04:50 AM   #3
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Re: Conjecture on cycle length and primes : prime abc conjec

Quote:
Originally Posted by CRGreathouse
That was me. I did my best to explain it and I voted to reopen the question after it had been closed initially.

But I still don't have much insight into what this cycle length is and without that I'm stumbling around in the dark.

That thanks!
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May 15th, 2013, 06:48 AM   #4
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Re: Conjecture on cycle length and primes : prime abc conjec

Thanks for CRG's rewrite, OEIS published a sequence generated by prime abc conjecture when c = 4 on May 15, 2013,see https://oeis.org/A225759.
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May 15th, 2013, 09:26 AM   #5
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Re: Conjecture on cycle length and primes : prime abc conjec

Congrats!
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May 15th, 2013, 05:35 PM   #6
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Re: Conjecture on cycle length and primes : prime abc conjec

Nice!
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