Mixing
Pattern in the primes and Goedels Incompleteness theorem
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Would finding a pattern in the primes affect goedels incompleteness theorem since his original proof has something to do with primes as axioms (also disclaimer, I don't actually know the proof but have a hand waving idea of what it's about). Would it for instance somehow invalidate the proof because if axioms could be systematically assigned a prime, would finding a pattern in the primes (or formula for them) be likened to finding one axiom for the set of all axioms?
Or am I thinking about this the wrong way and it would have no affect at all? Or more generally, are there any discoveries in prime numbers or number theory that we could make in the future that would affect the statement because of the math behind it?
prime-numbers
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
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show 2 more comments
$begingroup$
Would finding a pattern in the primes affect goedels incompleteness theorem since his original proof has something to do with primes as axioms (also disclaimer, I don't actually know the proof but have a hand waving idea of what it's about). Would it for instance somehow invalidate the proof because if axioms could be systematically assigned a prime, would finding a pattern in the primes (or formula for them) be likened to finding one axiom for the set of all axioms?
Or am I thinking about this the wrong way and it would have no affect at all? Or more generally, are there any discoveries in prime numbers or number theory that we could make in the future that would affect the statement because of the math behind it?
prime-numbers
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
$endgroup$
3
$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
2
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
1
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
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– Peter
Jan 5 at 8:47
1
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
1
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08
|
show 2 more comments
$begingroup$
Would finding a pattern in the primes affect goedels incompleteness theorem since his original proof has something to do with primes as axioms (also disclaimer, I don't actually know the proof but have a hand waving idea of what it's about). Would it for instance somehow invalidate the proof because if axioms could be systematically assigned a prime, would finding a pattern in the primes (or formula for them) be likened to finding one axiom for the set of all axioms?
Or am I thinking about this the wrong way and it would have no affect at all? Or more generally, are there any discoveries in prime numbers or number theory that we could make in the future that would affect the statement because of the math behind it?
prime-numbers
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
$endgroup$
Would finding a pattern in the primes affect goedels incompleteness theorem since his original proof has something to do with primes as axioms (also disclaimer, I don't actually know the proof but have a hand waving idea of what it's about). Would it for instance somehow invalidate the proof because if axioms could be systematically assigned a prime, would finding a pattern in the primes (or formula for them) be likened to finding one axiom for the set of all axioms?
Or am I thinking about this the wrong way and it would have no affect at all? Or more generally, are there any discoveries in prime numbers or number theory that we could make in the future that would affect the statement because of the math behind it?
prime-numbers
prime-numbers
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
asked Jan 5 at 8:01
Benjamin ThoburnBenjamin Thoburn
344112
344112
3
$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
2
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
1
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
$endgroup$
– Peter
Jan 5 at 8:47
1
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
1
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08
|
show 2 more comments
3
$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
2
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
1
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
$endgroup$
– Peter
Jan 5 at 8:47
1
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
1
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08
3
3
$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
2
2
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
1
1
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
$endgroup$
– Peter
Jan 5 at 8:47
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
$endgroup$
– Peter
Jan 5 at 8:47
1
1
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
1
1
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08
|
show 2 more comments
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$begingroup$
Why would it matter, necessarily? As far as I know, Godel's theorem makes no assumptions about order in the primes.
$endgroup$
– Eevee Trainer
Jan 5 at 8:05
2
$begingroup$
Regardless of finding a pattern in the primes, Godel’s theorem would still stand, it was rigorously proven, and is not some baseless assertion. The psuedo-random nature of the primes is merely an understandable example of something which may be unprovable.
$endgroup$
– Zachary Hunter
Jan 5 at 8:07
1
$begingroup$
No, a possible pattern of the primes has nothing to do with Goedels theorem. However, difficult number theoretical problems often have to do with primes, but not always. The Collatz conjecture for example has nothing to do with primes, but is extremely difficult to solve, if it can be solved at all.
$endgroup$
– Peter
Jan 5 at 8:47
1
$begingroup$
There are lots of patterns in primes. e.g. the use of Euler totient in the Legendre formula as an optimisation is a way of taking advantage of them.
$endgroup$
– Collag3n
Jan 5 at 13:18
1
$begingroup$
@Collag3n It depends what we understand under "pattern". I would consider a pattern to be a useful structure that would us allow to easily find huge primes. Such a pattern is not in sight. The patterns we know only hold for small primes. We know much about prime distributions, but when it comes to determine the primality of huge numbers, we soon arrive at limits. For example : We do NOT know whether infinite many primes of the form $n^2+1$ exist. We do NOT know whether infinite many twin primes exist. And we know no efficient method to find huge primes.
$endgroup$
– Peter
Jan 5 at 17:08