please empty your brain below
Strange fact: every prime number except 2 & 3 is either one more or one less than a multiple of 6

kev | 08.03.18 - 8:24 a.m. | #

@kev Not very strange, because it’s easy to see why. Primes other than 2 have to be odd, which means they’re all 1 less, 1 more or 3 more than a multiple of 6. But a number that’s 3 more than a multiple of 6 is a multiple of 3, so it can’t be prime (except for 3 itself).

Mike Scott | 08.03.18 - 8:27 a.m. | #

The most interesting primes to number theorists are probably Mersenne primes, primes that are one less than a power of two, such as 127 (2 ^ 7 - 1). 50 are known, of which the largest is 2 *^ 77,232,917 − 1, which has 23,249,425 digits. Whether or not there are infinitely many of them is an unsolved problem.

Mike Scott | 08.03.18 - 9:50 a.m. | #

There's a prime number with 6400 digits, of which 6399 are 9s. (Or so I've was lead to believe on Twitter the other day I confess I haven't tested it.) Not sure what group that fits into.

dg writes: That'd be a 'near-repunit prime'. Here.

Eric | 08.03.18 - 9:52 a.m. | #

Also rather neat are the pairs of primes, such as the "twin" prime (like 3 and 5, or 11 and 13) that differ by 2.

Or the "sexy" primes (like 5 and 11) which sound like they should be much more interesting: they differ by 6.

Andrew | 08.03.18 - 10:29 a.m. | #

I love the smell of nerdiness in the morning.

Jon Jones | 08.03.18 - 10:47 a.m. | #

My brain hurts! You lost me at "Prime Number"
Everything that follows is just cruel to us non-maffs types wot barely scraped an O level pass back in the dark ages!

Cornish Cockney | 08.03.18 - 11:01 a.m. | #

There is a prime number pencil (http://www.solipsys.co.uk/new/LeftTruncatablePrime.html) based on the largest left-truncatable prime, which indicates that DG is missing the final 7.

The point is that as you use and sharpen the pencil, you still have a prime number on it.

dg writes: Sorry for the truncation. Now fixed.

Michael Churchill | 08.03.18 - 12:35 p.m. | #

The only two-sided primes (over 100) are 313, 317, 373, 797, 3137, 3797 and 739397.

So is that the only known ones or has someone actually been able to prove that there can't be any more?

Pedantic of Purley | 08.03.18 - 12:37 p.m. | #

Pedantic - I was going to say that it will be the largest one. As there are a finite amount of left and right truncatable primes. (Simply keep adding a digit and if none of the 10 options give a prime number then you would not be able to extend the series, and by trial and error you would 'quickly' be able to show the full list).

Then I wondered about adding a 0 at the start and that messing up my logic, at which point I stopped thinking.

Messiah | 08.03.18 - 1:23 p.m. | #

I'd say that adding a leading zero does not create a new number - 07 or 007 or 000007 is just 7.

But according to Wolfram:
"There are exactly 83 right truncatable primes in base 10 ... There are exactly 4260 left truncatable primes in base 10 when the digit zero is not allowed ... If zeros are permitted, the sequence of left truncatable primes is infinite"

Obviously adding a trailing zero cannot create a prime.

Andrew | 08.03.18 - 4:15 p.m. | #

Back in the day I enjoyed reading about illegal primes.

Chris | Homepage | 08.03.18 - 5:25 p.m. | #

I shall enjoy my trips on the 337 that little bit more knowing that it's a member of a circular prime group.

Jonathan Wadman | 09.03.18 - 11:59 a.m. | #

The two-sided prime with most significance for dg on the day of this post is 53, I assume. Never underestimate the importance of stating the obvious.

JonathanC | 10.03.18 - 9:26 a.m. | #











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