# 7-5=2=5-3

Printable View

• Aug 6th 2010, 01:19 PM
Also sprach Zarathustra
7-5=2=5-3
$\displaystyle 3,5,7$ are primes.

And:

$\displaystyle 7-5=2=5-3$

Question:

Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)
• Aug 6th 2010, 01:56 PM
yeKciM
Quote:

Originally Posted by Also sprach Zarathustra
$\displaystyle 3,5,7$ are primes.

And:

$\displaystyle 7-5=2=5-3$

Question:

Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)

21-19=2=19-17

Edit : wrong , but i can't delete this (there need's to stay dumb things i'm writing so everyone can laugh) :D but now i'm thinking there is non another sequence like that ...
• Aug 6th 2010, 03:04 PM
Soroban
Hello, Also sprach Zarathustra!

Quote:

$\displaystyle 3,5,7$ are primes.

And: .$\displaystyle 7-5\:=\:2\:=\:5-3$

Are there any other triples of primes that differ by 2?

Since $\displaystyle 2$ is the only even prime,
. . any such triple of primes must be consecutive odd numbers.

It can be shown that, in a set of three consecutive odd numbers,
. . one of them will be divisible by 3.

Crank out a list of three consecutive odd numbers:

. . $\displaystyle \begin{array}{c}\rlap{/}1-3-5 \\ 3-5-7 \\ 5-7-\rlap{/}9 \\ 7-\rlap{/}9-11 \\ \rlap{/}9-11-13 \\ 11-13-\rlap{//}15 \\ 13-\rlap{//}15-17 \\ \vdots \end{array}$

And we see that $\displaystyle 3-5-7$ is the only such triple of primes.

• Aug 6th 2010, 03:05 PM
undefined
Quote:

Originally Posted by Also sprach Zarathustra
$\displaystyle 3,5,7$ are primes.

And:

$\displaystyle 7-5=2=5-3$

Question:

Are there any triples of primes so the difference between any two of sequential number equals 2? (Except the above...)

No because for {p, p+2, p+4}, one of them must be divisible by 3, obviously p=2 does not work, and for p>3 divisibility by 3 will mean composite.

Edit: Soroban beat me to it!
• Aug 6th 2010, 03:17 PM
Also sprach Zarathustra
1 point to Soroban and 1 point to undefined.
• Aug 7th 2010, 06:30 AM
wonderboy1953
Twin primes
Just to add my two cents worth in, there may be an infinity of twin primes however (Calvin Clawson's [I]Math Mysteries[I] is worthwhile reading).
• Aug 7th 2010, 09:17 AM
Wilmer
251-257-263-269

4 consecutive primes, any 2 consecutive have same difference (6).
• Aug 7th 2010, 10:55 AM
undefined
Quote:

Originally Posted by Wilmer
251-257-263-269

4 consecutive primes, any 2 consecutive have same difference (6).

Along those lines,

Green-Tao Theorem
• Aug 7th 2010, 12:36 PM
Wilmer
Quote:

Originally Posted by undefined
Along those lines,
Green-Tao Theorem

YIKES!! They're up to 26 consecutives; bunch of nuts !!
• Aug 7th 2010, 07:14 PM
undefined
Quote:

Originally Posted by Wilmer
YIKES!! They're up to 26 consecutives; bunch of nuts !!

:D More nutjob info here

Primes in arithmetic progression - Wikipedia, the free encyclopedia