1. ## 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...)

2. 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) but now i'm thinking there is non another sequence like that ...

3. Hello, Also sprach Zarathustra!

$\displaystyle 3,5,7$ 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.

4. 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!

5. 1 point to Soroban and 1 point to undefined.

6. ## 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).

7. 251-257-263-269

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

8. Originally Posted by Wilmer
251-257-263-269

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

Green-Tao Theorem

9. Originally Posted by undefined
Along those lines,
Green-Tao Theorem
YIKES!! They're up to 26 consecutives; bunch of nuts !!

10. Originally Posted by Wilmer
YIKES!! They're up to 26 consecutives; bunch of nuts !!
More nutjob info here

Primes in arithmetic progression - Wikipedia, the free encyclopedia