Problems & Puzzles: Puzzles

 Puzzle 791. Inserting consecutive primes This puzzle is a variation of another posted by Claudio Meller as the entry 1398 in his always interesting site. Here we ask for these numbers N that result prime numbers after inserting the first consecutive primes, one at a a time, from 2 to P, except the prime "3" in successive positions in N, from left to right. Example: N=333 is a solution because 2333, 3533, 3373 and 33311 are prime numbers. As a matter of fact, below 2^32 I have found 21 of these asked numbers N, being the smallest "27" and largest "2653294371". Q. Find more solutions.

At last, one more solution!...

Jan van Delden wrote on May 1, 2018:

I found the following 21 values below 2^32 (and 10^10):

Digits 2: 27,33,39,57,93

Digits 3: 333

Digits 4: 3747,5073,5997,7239

Digits 5: 10053,22419
Digits 6: 349731,425991,714807

Digits 7: 1719279

Digits 8: 81453303

Digits 9: 406253439,481683189,886662423

Digits 10: 2653294371

I also found the following NEW solution:

Digits 11: 61605403707

Digits 12: No solution

Let’s call the number we seek N and the result after insertion (possibly at the end) with prime p: M.
It is possible to link N,M and p modulo 11. This is rather easy if p has two digits.

Suppose N=XY and M=XpY and p has two digits.

1. Y has an even number of digits, 2k. (possibly 0, when we “insert” p at the end).
N = X*10^(2k)+Y, M=X*10^(2k+2)+p*10^(2k)+Y.  We use: 100=1 mod 11.
We now have  M mod 11 = X*10^(2k)*100+p*100^k+Y mod 11 = X*10^(2k)+p+Y mod 11 = N mod 11 + p mod 11.
2. Y has an odd number of digits, 2k+1. (I choose the +sign for convenience).
N = X*10^(2k+1)+Y, M=X*10^(2k+3)+p*10^(2k+1)+Y. We use: 100=1 mod 11 and 10=-1 mod 11.
We have M mod 11 = X*10^(2k+1)*100 +p*10*100^k+Y mod 11 = X*10^(2k+1) -p + Y mod 11 = N mod 11 – p mod 11.

For 2-digit primes p inserted at “even positions” we have M=N+p mod 11 (A).
For 2-digit primes p inserted at “odd positions” we have M=N-p mod 11 (B).

Since we search for prime values for M we should have that M<>0 mod 11, or:
N <> -p mod 11 (p inserted at “even positions”)
N <> p mod 11 (p inserted at “odd positions”).

If we want to compute a solution N having 12 digits the primes to insert are {2,5,7,11,13,17,19,23,29,31,37,41,43}.

The primes p with 2 digits at the “even positions” are {13,19,29,37,43}, so -p mod 11 in {9,3,4,7,1}.
The primes p with 2 digits at the “odd position” are {11,17,23,31,41}, so p mod 11 in {0,6,1,9,8}.

Our N may not be in the set {0,1,3,4,6,7,8,9} mod 11, so N mod 11 must be in {2,5,10}.

If a solution exists N has 21 digits or less.

The first prime p for which there is no free residue is 89 (prime number 24).
There is also no free residue for prime 97 (prime number 25).

If N has k digits the largest prime used is prime number k+2. Thus there are no solutions if N has 22 digits or more.

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