TIL...... [modified]
-
...an extraordinary fact - probably known to half of you, but nevertheless I thought I'd share it. It's not an alogorithm, but never mind! The square of any prime number >= 5 can be expressed as 1 + a multiple of 24 or
If p is prime, then p ^ 2 = 1 + 24 * n, for some nDon't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!) [edit] When I ask "where does 24 come from?" I am not asking for a mathermatical proof - I can look that up myself. It's more just a metaphysical musing...)modified on Thursday, November 4, 2010 6:37 AM
-
Try
p ^ 2 = 1 + 24 * (p - 6)
Just say 'NO' to evaluated arguments for diadic functions! Ash
OK...
p^2 - 24*p + 143 = 0So..? -
OK...
p^2 - 24*p + 143 = 0So..? -
Me :confused: too - ome of us is being a bit dumb here (most likely me...) I just re-wrote the equation you posted... what does it prove? Is there a typo in it (yours)?
-
OK...
p^2 - 24*p + 143 = 0So..?a quadratic equation like that has at most two solutions, so it is hardly a way to discover lots of primes. :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
modified on Thursday, November 4, 2010 12:05 PM
-
...an extraordinary fact - probably known to half of you, but nevertheless I thought I'd share it. It's not an alogorithm, but never mind! The square of any prime number >= 5 can be expressed as 1 + a multiple of 24 or
If p is prime, then p ^ 2 = 1 + 24 * n, for some nDon't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!) [edit] When I ask "where does 24 come from?" I am not asking for a mathermatical proof - I can look that up myself. It's more just a metaphysical musing...)modified on Thursday, November 4, 2010 6:37 AM
that is correct, even when n needs to be 1/8 for p=2. :-D
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
-
a quadratic equation like that has at most two solutions, so it is hardly a way to discover lots of primes. :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
modified on Thursday, November 4, 2010 12:05 PM
We have a saying round here: "Don't tell I, tell 'e"... especially as that particular one doesn't even have any (real) solutions at all!
-
that is correct, even when n needs to be 1/8 for p=2. :-D
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
Actually, that's QI.. it also "works" for p=3 (n=1/3) - so in fact, the *only* value of p for which n is neither an integer nor a proper fraction is 4.[edit] OOPS what an idiot! since when was 4 a prime! :-O
-
Me :confused: too - ome of us is being a bit dumb here (most likely me...) I just re-wrote the equation you posted... what does it prove? Is there a typo in it (yours)?
-
We have a saying round here: "Don't tell I, tell 'e"... especially as that particular one doesn't even have any (real) solutions at all!
you'll have more luck with
x2 + x + 41
for natural values of x. :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
-
No typo as far as I am aware. I just thought you wanted to know what was the value of 'n' in your original question.
Just say 'NO' to evaluated arguments for diadic functions! Ash
Ah, got you - except that it doesn't work for all values of p p=5, n=1 != p-6 p=7, n=2 != p-6 p=11, n=5 ok p=13, n=7 ok p=17, n=12 != p-6 ...
-
you'll have more luck with
x2 + x + 41
for natural values of x. :)
Luc Pattyn [Forum Guidelines] [Why QA sucks] [My Articles] Nil Volentibus Arduum
Please use <PRE> tags for code snippets, they preserve indentation, and improve readability.
Now you're just confusing me....
-
...an extraordinary fact - probably known to half of you, but nevertheless I thought I'd share it. It's not an alogorithm, but never mind! The square of any prime number >= 5 can be expressed as 1 + a multiple of 24 or
If p is prime, then p ^ 2 = 1 + 24 * n, for some nDon't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!) [edit] When I ask "where does 24 come from?" I am not asking for a mathermatical proof - I can look that up myself. It's more just a metaphysical musing...)modified on Thursday, November 4, 2010 6:37 AM
Well, I know you are not asking for a mathematical proof, but like I am much better at maths than metaphisics... Where does 24 come from? Since p is prime and greater than 5, p cannot be an even number. So: p2-1=(p-1)*(p+1). Like p is odd, both (p-1) and (p+1) are even, so we can say: p2-1=2a*2b=4ab. Like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 4, so 2a is multiple of 4 or 2b is multiple of 4, so we can express this like: p2-1=4a*2c=8ac or p2-1=4b*2d=8bd I will examine just one of these two cases becouse they are symmetrical. Now, like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 3, so: p2-1=8a*3d=24ad or p2-1=8c*3e=24ce And there it is.
-
Well, I know you are not asking for a mathematical proof, but like I am much better at maths than metaphisics... Where does 24 come from? Since p is prime and greater than 5, p cannot be an even number. So: p2-1=(p-1)*(p+1). Like p is odd, both (p-1) and (p+1) are even, so we can say: p2-1=2a*2b=4ab. Like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 4, so 2a is multiple of 4 or 2b is multiple of 4, so we can express this like: p2-1=4a*2c=8ac or p2-1=4b*2d=8bd I will examine just one of these two cases becouse they are symmetrical. Now, like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 3, so: p2-1=8a*3d=24ad or p2-1=8c*3e=24ce And there it is.
You haven't explained the last step correctly, and the statement
_Erik_ wrote:
Now, like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 3
is wrong.
-
You haven't explained the last step correctly, and the statement
_Erik_ wrote:
Now, like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 3
is wrong.
It is not wrong, becouse p is a prime number, what means: if p mod 3 = 1 then (p-1) mod 3=0, so (p-1) is multiple of 3. if p mod 3 = 2 then (p+1) mod 3=0, so (p+1) is multiple of 3. if p mod 3 = 0 then p would not be prime and we would not be following our premise
-
It is not wrong, becouse p is a prime number, what means: if p mod 3 = 1 then (p-1) mod 3=0, so (p-1) is multiple of 3. if p mod 3 = 2 then (p+1) mod 3=0, so (p+1) is multiple of 3. if p mod 3 = 0 then p would not be prime and we would not be following our premise
Yes, sorry - I was forgetting that p is prime! Time to knock off for the day I think...
-
Well, I know you are not asking for a mathematical proof, but like I am much better at maths than metaphisics... Where does 24 come from? Since p is prime and greater than 5, p cannot be an even number. So: p2-1=(p-1)*(p+1). Like p is odd, both (p-1) and (p+1) are even, so we can say: p2-1=2a*2b=4ab. Like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 4, so 2a is multiple of 4 or 2b is multiple of 4, so we can express this like: p2-1=4a*2c=8ac or p2-1=4b*2d=8bd I will examine just one of these two cases becouse they are symmetrical. Now, like (p-1) and (p+1) are two consecutive even numbers, one of them must be multiple of 3, so: p2-1=8a*3d=24ad or p2-1=8c*3e=24ce And there it is.
-
Try
p ^ 2 = 1 + 24 * (p - 6)
Just say 'NO' to evaluated arguments for diadic functions! Ash
-
...an extraordinary fact - probably known to half of you, but nevertheless I thought I'd share it. It's not an alogorithm, but never mind! The square of any prime number >= 5 can be expressed as 1 + a multiple of 24 or
If p is prime, then p ^ 2 = 1 + 24 * n, for some nDon't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!) [edit] When I ask "where does 24 come from?" I am not asking for a mathermatical proof - I can look that up myself. It's more just a metaphysical musing...)modified on Thursday, November 4, 2010 6:37 AM
-
...an extraordinary fact - probably known to half of you, but nevertheless I thought I'd share it. It's not an alogorithm, but never mind! The square of any prime number >= 5 can be expressed as 1 + a multiple of 24 or
If p is prime, then p ^ 2 = 1 + 24 * n, for some nDon't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!) [edit] When I ask "where does 24 come from?" I am not asking for a mathermatical proof - I can look that up myself. It's more just a metaphysical musing...)modified on Thursday, November 4, 2010 6:37 AM
For p = 5, p^2 = 24 * 1 + 1 All primes > 5 have the form p = 30*k + l, where l is 1, 7, 11, 13, 17, 19, 23 or 29. All other values of l have a common factor > 1 with 30, and therefore cannot produce primes. p^2 = 900*k^2 + 60*k*l + l^2 Rewrite the first 2 terms as 60*( 15*k^2 + k*l ). The sum in the parentheses is always even, so the entire expression is divisible by 24. We are left with l^2. By inspection: 1^2 = 24 * 0 + 1 7^2 = 24 * 2 + 1 11^2 = 24 * 5 + 1 13^2 = 24 * 7 + 1 17^2 = 24 * 12 + 1 19^2 = 24 * 15 + 1 23^2 = 24 * 22 + 1 29^2 = 24 * 35 + 1 QED