Problems & Puzzles: Puzzles
Puzzle 240. Consecutive numbers and consecutive prime factors
Here we will ask for pairs of consecutive numbers (n, n+1) such that each is composed only by consecutive prime factors.
As a matter of fact I have found four examples:
1. Can you find the next 3 examples?
2. Can you find a moderate large example, let's say 20 digits?
3. Can you find a pair of these where the primes involved are not repeated in each number?
Jon Wharf, Adam Stinchcombe, Enoch Haga, Faride Firoozbakht, Eric Brier and J. K. Andersen solved Q1.
Wharf, Firoozbakht, Brier and Andersen observed the relation of these kind of asked pairs and the twin primes as a basis to construct a smart approach in order to get larger examples and, consequently, solved Q2 by far.
Nobody has found an example for the case Q3, so probably there is not any one.
Q1: Here are the first pairs below 100000, as reported by several of the solvers of this puzzle.
* given as examples in
Q2: What is the relation on the pairs (n, n+1) and the twin primes, mentioned above?
Let p and p+2 twin primes; so if n=p(p+2) then n+1=(p+1)^2. From start n is a product of consecutive primes (because p & p+2 are twin and necessarily consecutive primes), so all we need is that p+1 (an even and composite quantity) can be factorized as a product of consecutive primes.
While not all the known solutions are of this type (all pairs in the list above marked with an asterisk are not of this type), this model provides a basis in order to hunt systematically the asked (n , n+1) pairs of large size for this puzzle
Before presenting the large solutions gotten I would like to say that Wharf and Andersen neatly noticed that after (4374, 4375) they are all of the 'twin-prime + square' type (Why?... See at the bottom the new questions emerging of this)
Large solutions: In order to get large solutions you should use the twin-prime + square model. In order to guarantee that (p+1) is a product of consecutive primes you have several options as: p=k*q# -1 or p=k-q! - 1, and so on, all related with the 'minus one" primorial or factorial functions, using a k multiplier appropriate, that is to say a multiplier k having its largest prime factor less that q.
J. K. Andersen sent the largest one (5138 digits)
Others sent by Andersen (I checked Chris Caldwell's prime database and found two large twin primes x+/-1 satisfying the requirement for x) were:
2846!4+/-1 with 2151 digits was found in 1992 by Harvey Dubner. 2846!4 is the multifactorial 2846*2842*2838*...*2 which contains all prime factors <= 1423
5775*2^5907+/-1 with 1782 digits was found in 1998 by Bradford Brown with Yves Gallot's Proth.exe. 5775 = 3*5^2*7*11 which is fine for our purpose.
Faride sent the following one:
Jon Wharf sent this one:
Adam Stinchcombe wrote:
Q3. In order to get this target you should use the twin-pair + square model together with the primorial minus one option. This is what two solvers wrote: