
022812, 07:52 AM #21
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Captain Kremmen, U have seen the wrong thread in ask nrich it is actually "Is it known" of Ask nrich site not "Are these facts known" thread of ask nrich site?
regards.Last edited by indianmath; 022812 at 07:57 AM. Reason: added information

022812, 08:04 AM #22
I can't see how it's a reformulation of Golbach's conjecture.
If he wants to say that, he needs to show how you derive one from the other.
I'll look at your link later.
How about this as a wording for the conjecture?
"Any odd number which is not a prime, except the number nine, is the sum of all its prime factors, plus other primes.
In this summation no number is used more than once."Last edited by Captain Kremmen; 022812 at 08:11 AM.

022812, 08:08 AM #23
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022812, 08:14 AM #24
I hope this isn't of use to codebreakers, otherwise you'll have the CIA or its equivalent knocking on your door.

022812, 08:16 AM #25
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022812, 08:22 AM #26
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022812, 10:30 AM #27
indianmath, your comments about being able to prove this are ambiguous, particularly at the NRICH link. Can you prove this mathematically or is it a pure conjecture at this point?

022812, 11:34 AM #28

022812, 12:05 PM #29
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RJBeery, If Bertrand's theorem is true then my theorem is also true as u see after 9 the next number which comes into my theorem's context is 15 which i have shown in the statement & if u want to extend it more i.e for any number then except for 2,6 & 9 u will find no exception of my theorem.
regards.Last edited by indianmath; 022812 at 12:16 PM. Reason: corrected spelling

022812, 12:14 PM #30
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022812, 02:44 PM #31
Ah, that was the piece I needed.
Shorter, stronger claim:
"All natural numbers other than 1, 4, 6, 8, 9, or 22 can be partitioned into unique primes such that all of their prime factors are present in the partition."
1 can't be partitioned into primes
Any prime can only be partitioned one way into primes including its factors, i.e., into itself
4, 6, 8, 9, 22 can't be partitioned into unique primes once their prime factors are accounted for.
Here is an (updated) list of how some small composite numbers could be partitioned:
10 = ( 2 + 5 ) + ( 3 )
12 = ( 2 + 3 ) + ( 7 )
14 = ( 2 + 7 ) + ( 5 )
15 = ( 3 + 5 ) + ( 7 )
16 = ( 2 ) + ( 3 + 11 )
18 = ( 2 + 3 ) + ( 13 )
20 = ( 2 + 5 ) + ( 13 )
21 = ( 3 + 7 ) + ( 11 )
24 = ( 2 + 3 ) + ( 19 )
25 = ( 5 ) + ( 3 + 17 )
26 = ( 2 + 13 ) + ( 11 )
27 = ( 3 ) + ( 5 + 19 )
28 = ( 2 + 7 ) + ( 19 )
30 = ( 2 + 3 + 5 ) + ( 7 + 13 )
A more interesting question is, for each natural number, how many distinct partitions into unique primes that include all prime factors there are. Note that zero can be partitioned into the empty set, so it has more solutions than 1.Last edited by rpenner; 022812 at 03:04 PM. Reason: Found a mistake

022812, 03:16 PM #32
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022912, 12:23 AM #33
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 104
Since the above is true then these are also true
1)If n be any even integer & the summation of its prime factors is odd then the odd number remaining would be a prime number &/or if not prime then if the above exceptions are subtracted from it then with minimum two exceptions that remaining odd number would give a prime number.
2)If n be any odd integer & the summation of its prime factors is even then the odd number remaining would be a prime number &/or if not prime then if the above exceptions are subtracted from it then with minimum two exceptions that remaining odd number would give a prime number.
regards.

022912, 12:46 AM #34
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Any counter example to the above statements?
regards.

022912, 10:05 AM #35
indianmath, your last two statements have an "ad hoc" flavor to them; I get the impression that you made them after checking the first few dozen elements or so..?
I used to study number theory a bit and something that I didn't appreciate were conjectures that seemed to rely on statistical tendencies, as opposed to resting squarely upon deeper, hidden conceptual foundations.
Take Goldbach's Conjecture, for example: To find a counterexample we would have to produce a number n such that (x) and (nx) are both composite for all x < n/2. The Prime Number Theorem states that there are roughly
primes less than n. The "blind odds" of either (x) or (nx) being composite are then
and therefore the "blind odds" of finding a counterexample to Goldbach's Conjecture for all x < n/2 are roughly
At n = 100, this number is only 9% and at 10000 this number drops to ! Therefore, Goldbach's Conjecture could very well be true solely due to statistical curiosities rather than being indicative of any fundamental law...

022912, 12:01 PM #36
I agree that indianmath is overconfident that there are no counterexamples.
To do that, he would need to find a proof.
Nevertheless, noone has yet shown that his idea is not new, or that it is wrong. So it is very interesting.
@Indianmath
I get the feeling for some reason that you are a young person still in education.
What stage of education are you at?

022912, 12:16 PM #37
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I'm not overconfident I'm saying it confidently that since all the above things are true therefore
" Any natural number except (4, 6 & certain prime numbers ) can be expressed as summations of at least Two distinct primes".
I think this statement proves Goldbach's conjecture also side by side.
regards.Last edited by indianmath; 022912 at 12:22 PM. Reason: found an error

022912, 02:07 PM #38
Have you got a mathematical proof for that.
So far you have asked for exceptions.
As you know, that is not a proof.
You seem very confident that it will never fail at a higher number.
What gives you that confidence?
The next number it fails at could be very high.
Up to what number have you checked?
Or have you got a proof that it will never fail?Last edited by Captain Kremmen; 022912 at 02:26 PM.

030112, 05:18 AM #39
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030112, 07:10 AM #40
That's fine. Let's not argue.
I could be wrong saying that you need to provide a mathematical proof in order to say there are no further counterexamples.
Or perhaps you already have, and I didn't catch it.
I'll leave it to Rpenner.
He understands this stuff far better than I do.
If he doesn't think there is a problem, there probably isn't.
Personally, I would wonder why there are exceptions at 9 and 22, and then no further exceptions.
I would expect more. Not sure why.
Even if there were more exceptions, that would not make your idea less interesting.
It would provide a new series of numbers for people to ponder on.Last edited by Captain Kremmen; 030112 at 08:50 AM.
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