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A prime triplet
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Risto Kauppila
Mar 27, 2012, 12:57:08 PM3/27/12
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The set of three primes, { 2, 3, 13 }, has the following property.
2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
|2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
are all primes.
I found no other such triplet among the first 10000 primes.
Can you find one or prove the non-existence of them?
Perhaps a mere coincidence, but 2*3*13 = 78 = gcd(W1-1, W2-1),
where W1 = 1093 and W2 = 3511 are the only known Wieferich
primes.
Rike
2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
|2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
are all primes.
I found no other such triplet among the first 10000 primes.
Can you find one or prove the non-existence of them?
Perhaps a mere coincidence, but 2*3*13 = 78 = gcd(W1-1, W2-1),
where W1 = 1093 and W2 = 3511 are the only known Wieferich
primes.
Rike
Mensanator
Apr 3, 2012, 12:02:16 AM4/3/12
to
On Mar 27, 11:57 am, Risto Kauppila
product of 2 primes
added to a third prime will always be EVEN, thus, not prime.
(excluding 2)
> Can you find one or prove the non-existence of them?
QED
<risto.kauppi...@NOSPAM.saunalahti.fi.invalid> wrote:
> The set of three primes, { 2, 3, 13 }, has the following property.
> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
> are all primes.
> I found no other such triplet among the first 10000 primes.
Why did you search through 10000 primes? Don't you realize that the
> The set of three primes, { 2, 3, 13 }, has the following property.
> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
> are all primes.
> I found no other such triplet among the first 10000 primes.
product of 2 primes
added to a third prime will always be EVEN, thus, not prime.
(excluding 2)
> Can you find one or prove the non-existence of them?
Risto Kauppila
Apr 3, 2012, 3:17:34 AM4/3/12
to
03.04.2012 07:02, Mensanator kirjoitti:
> On Mar 27, 11:57 am, Risto Kauppila
> <risto.kauppi...@NOSPAM.saunalahti.fi.invalid> wrote:
>> The set of three primes, { 2, 3, 13 }, has the following property.
>> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
>> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
>> are all primes.
>> I found no other such triplet among the first 10000 primes.
>
> Why did you search through 10000 primes? Don't you realize that the
> product of 2 primes
> added to a third prime will always be EVEN, thus, not prime.
> (excluding 2)
>
??
> On Mar 27, 11:57 am, Risto Kauppila
> <risto.kauppi...@NOSPAM.saunalahti.fi.invalid> wrote:
>> The set of three primes, { 2, 3, 13 }, has the following property.
>> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
>> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
>> are all primes.
>> I found no other such triplet among the first 10000 primes.
>
> Why did you search through 10000 primes? Don't you realize that the
> product of 2 primes
> added to a third prime will always be EVEN, thus, not prime.
> (excluding 2)
>
Of course such triplets are of the form
{ 2, p, q }, p and q odd primes.
Risto Kauppila
Apr 7, 2012, 4:39:49 AM4/7/12
to
27.03.2012 19:57, Risto Kauppila kirjoitti:
> The set of three primes, { 2, 3, 13 }, has the following property.
> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
> are all primes.
> I found no other such triplet among the first 10000 primes.
> Can you find one or prove the non-existence of them?
>
Since 2 must be one of the primes, I'm actually trying to find
> The set of three primes, { 2, 3, 13 }, has the following property.
> 2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
> |2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
> are all primes.
> I found no other such triplet among the first 10000 primes.
> Can you find one or prove the non-existence of them?
>
pairs of odd primes (p, q) satisfying the above property.
More primes from { 2, 3, 13 }:
(2 + 3*13) + (3 + 2*13) + (13 + 2*3) = 89
(3*13 -2) + (2*13 - 3) + (13 - 2*3) = 67
2*3 + 2*13 + 3*13 = 71
3*13 + 2*13 - 2*3 = 59
3*13 - 2*13 + 2*3 = 19
3*13 - 2*13 - 2*3 = 7
Henning F. Jensen
Apr 9, 2012, 5:49:56 AM4/9/12
to
>"Risto Kauppila" skrev i meddelelsen
>news:p3Tfr.28644$I33....@uutiset.elisa.fi...
2 + 3*13 = 41, 3 + 2*13 = 29, 13 + 2*3 = 19,
|2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
Sketch of proof:
|2 - 3*13| = 37, |3 - 2*13| = 23, |13 - 2*3| = 7
Consider the triple (2, 6q - 1, p), (q and p integers 6q-1<=p)
If (p mod 3) =1 or (p mod 3) =2 then at least one of the 6 expressions is a
multiple of 3
Consider the triple (2, 6q +1, p), (q and p integers 6q+1<=p)
If (p mod 3) =1 or (p mod 3) =2 then at least one of the 6 expressions is a
multiple of 3
So if all 6 expressions are primes then p is a multiple of 3 and hence not a
prime
Consider the triple (2, 3, p), (p integer, p>13)
If (p mod 7) = 1,2,3,4,5,6 then at least one of the 6 expression is a
multiple of 7
So if all 6 expressions are primes then p is a multiple of 7 and hence not
prime.
This leaves triples (2,3,p) p=3,5,7,11,13 of which p=13 is the only one
where all 6 expressions are primes.
Henning
Risto Kauppila
Apr 10, 2012, 4:38:16 AM4/10/12
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