Prime 479

[enzocreti]

Is 479 the only example of prime p(k) such that reversed is 2*p(k+1)?

974=2*487 infact

I did not found another example...is this due to the rarefaction of primes?

[CRGreathouse]

It's hard to guess, but I would not be surprised if there were infinitely many.

[enzocreti]

Originally Posted by CRGreathouse

It's hard to guess, but I would not be surprised if there were infinitely many.

I searched up to 10^10 but no other example

[Dr Sardonicus]

Originally Posted by enzocreti

Is 479 the only example of prime p(k) such that reversed is 2*p(k+1)?

974=2*487 infact

I did not found another example...is this due to the rarefaction of primes?

I wonder what a little elementary number theory might reveal...

The most significant digit (MSD) d of p_k would have to be even, so d = 2, 4, 6, or 8. Since d < 9, we can say that the MSD of p_k+1 is either d or d+1. If d > 4, 2*p_k+1 would have more digits than p_k, so the only possibilities are d = 2 or d = 4.

If 2*p_k+1 == 2 (mod 10), then p_k+1 == 1 (mod 5). If 2*p_k+1 == 4 (mod 10), then p_k+1 == 2 (mod 5).

So, either p_k has MSD d = 2, and p_k+1 == 1 (mod 10), or p_k has MSD d = 4, and p_k+1 == 7 (mod 10).

These conditions (especially the MSD being either 2 or 4) narrow things down a bit...

[axn]

Originally Posted by Dr Sardonicus

These conditions (especially the MSD being either 2 or 4) narrow things down a bit...

2...1 will not work, because 2...1 x 2 = 4...2 or 5...2, and neither reversed (2..4 or 2..5) yields a prime.

Only 4...7 will work. Here too, it will have to be 4[579]...7
If 2nd digit < 5, doubling will leave the MSD as 8, which means the reverse can't be a prime, therefore 2nd digit >= 5.
Also 4...7 x 2 = 9...[odd]4. Since reverse (4[odd]...9) and 4...7) are consecutive primes, they'll share the second digit (this is not hard and fast rule, but once the numbers get big enough this will hold).

[enzocreti]

do you guess that below 10^14 there could be another prime of this type?

[Dr Sardonicus]

As an exercise in writing Pari scripts, I told Pari to look at numbers of the form 45...9, 47...9 and 49...9 up to the limit 50 million, which was as far as I could go and stay under the primelimit I had set. If the number formed by reversing the decimal digits was between 2 and 2.05 times the original number, I had it check whether the original number and half the reversed number were both (pseudo)prime. If they were, I told it to find the difference in prime counts between the original and half the reversed number. If this difference was less than 100, I told Pari to print the original and reversed numbers, and the difference in prime counts.

I'm not sure what (if anything) the results indicate.

45319 91354 32
458819 918854 51
4547909 9097454 48
4597919 9197954 69

479 974 1
4759 9574 2
47459 95474 28
47659 95674 17
47869 96874 48
47969 96974 44
4797959 9597974 65
47025049 94052074 67
47387749 94778374 86
47525059 95052574 68

49199 99194 37
49499 99494 23
495199 991594 55
4946989 9896494 70
4969399 9939694 28
49199389 98399194 14

[enzocreti]

Originally Posted by Dr Sardonicus

As an exercise in writing Pari scripts, I told Pari to look at numbers of the form 45...9, 47...9 and 49...9 up to the limit 50 million, which was as far as I could go and stay under the primelimit I had set. If the number formed by reversing the decimal digits was between 2 and 2.05 times the original number, I had it check whether the original number and half the reversed number were both (pseudo)prime. If they were, I told it to find the difference in prime counts between the original and half the reversed number. If this difference was less than 100, I told Pari to print the original and reversed numbers, and the difference in prime counts.

I'm not sure what (if anything) the results indicate.

45319 91354 32
458819 918854 51
4547909 9097454 48
4597919 9197954 69

479 974 1
4759 9574 2
47459 95474 28
47659 95674 17
47869 96874 48
47969 96974 44
4797959 9597974 65
47025049 94052074 67
47387749 94778374 86
47525059 95052574 68

49199 99194 37
49499 99494 23
495199 991594 55
4946989 9896494 70
4969399 9939694 28
49199389 98399194 14

Ok no prime below 10^10

[CRGreathouse]

Originally Posted by enzocreti

do you guess that below 10^14 there could be another prime of this type?

Highly unlikely. If I had to guess I'd say a few % chance you'd find one below 10^50. To check more carefully you'd want to look at the contributions of the major primes and then calculate by digit. The most important primes here are those dividing b(b-1) with b = 10 since you're using decimal, so 2, 3, and 5.

[enzocreti]

Originally Posted by CRGreathouse

Highly unlikely. If I had to guess I'd say a few % chance you'd find one below 10^50. To check more carefully you'd want to look at the contributions of the major primes and then calculate by digit. The most important primes here are those dividing b(b-1) with b = 10 since you're using decimal, so 2, 3, and 5.

There is no program that can reach 10^50 I think

[CRGreathouse]

Originally Posted by enzocreti

There is no program that can reach 10^50 I think

I suspect not. 10^25, probably. 10^35, only with great effort unless there are more features than meet the eye. But not 10^50.