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29 Mar 2012 10:38:59 IST

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Do any odd perfect numbers exist?

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boywholived

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29 Mar 2012 11:37:13 IST

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No odd perfect numbers is found till now .But there are no proofs stating the non existence of perfect numbers. Many great mathematicians said they won't exist. I have read somewhere that with the help of high performance supercomputers the perfect numbers were calculated upto very large numbers and weren't found odd. But nobody rules out the probability of finding an odd perfect number

sathyaram

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29 Mar 2012 12:12:43 IST

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This Might help u machan ......

Odd perfect numbers

Unsolved problems in mathematics
Are there any odd perfect numbers?

It is unknown whether there are any odd perfect numbers, though various results have been obtained. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. All perfect numbers are also Ore's harmonic numbers, and it has been conjectured as well that there are no odd Ore's harmonic numbers other than 1.

Any odd perfect number N must satisfy the following conditions:

N > 10¹⁵⁰⁰, result published in 2012.
N is of the form

$N=q^{\alpha} p_1^{2e_1} \cdots p_k^{2e_k},$

where:

q, p₁, ..., p_k are distinct primes (Euler).
q ≡ α ≡ 1 (mod 4) (Euler).
The smallest prime factor of N is less than (2k + 8) / 3.
Either q^α > 10⁶², or p_j^2e_j > 10⁶² for some j.
N < 2^{4^k+1}.^[11]

The largest prime factor of N is greater than 10⁸.
The second largest prime factor is greater than 10⁴, and the third largest prime factor is greater than 100.^[13]^[14]
N has at least 101 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.

In 1888, Sylvester stated:

...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.

srinivasan

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29 Mar 2012 13:26:44 IST

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thank you both of you :) i was thinking about it for long time :) great information sathyaram macha :) i thought only 6 millinieum problems are unsolved in mathematics :)

srinivasan

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29 Mar 2012 13:29:30 IST

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another question - how to rate people ? i dont understnad it

sathyaram

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29 Mar 2012 13:38:36 IST

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rate is nthin but to like it da machan..... For Ex.Just click Like for this comment...That is rate...HAhahah lolz :P

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