Problems & Puzzles: Puzzles

 Puzzle 970. A309566 Daniel Suteu sent the following nice puzzle. Q. For A309566, find a(5) or, at least, an upper-bound for a(5). a(n) is the least prime that can be written as a sequence of primes separated by n single zeros, and where every 0-splitting is prime. The only four terms currently known, are: 307, 130307, 309370307, 30281172370306703. For example, 130307 is a term because 3, 7, 13, 307, 1303, 130307 are all prime. The n-th term contains binomial(n+2,n) zero-splitting primes (including itself), not necessarily distinct. The 10 zero-splitting primes in 309370307 are: 3, 3, 7, 307, 937, 30937, 93703, 3093703, 9370307, 309370307. The 15 zero-splitting primes in 30281172370306703 are: 3, 3, 3, 67, 3067, 6703, 306703, 28117237, 2811723703, 3028117237, 302811723703, 2811723703067, 281172370306703, 302811723703067, 30281172370306703.

Contributions came from Oscar Volpatti, Emmanuel Vantieghem

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Oscar wrote on Set 14, 2019:

Sequence A309566 can't be extended beyond a(4).

We are asked to find some number B = [b(0),0,b(1),0,...,0,b(n)] such that:
1) each block b(k) is a prime with a zero-less base-10 representation;
2) each 0-splitting involving two or more blocks (including B itself) is a prime too.

No such number exists for n>=5, so sequence A309566 can't be extended beyond a(4).
More precisely, property 2 fails for n>4 because some 0-splitting admits 3 as a proper divisor.

Compute the n+1 residues c(i) = b(i) % 3.
Let x be the number of occurrences of the residue 0, y be the number of occurrences of non-zero residues, with x+y = n+1.

Suppose that equality c(i) = c(i+1) = 0 holds for some index i; then 3 divides the 0-splitting [b(i),0,b(i+1)].
Therefore, any two occurrences of the residue 0 must be separated by some non-zero residue, and inequality x<=y+1 must hold.
As x+y = n+1, we can rewrite it in terms of n and y as (n+1-y)<=y+1, or simply n<=2y.

Suppose that inequality y>=3 holds, and let i,j,k be the indexes of the first such non-zero residues, with c(j) = r.
if c(i) = 3-r, then 3 divides the 0-splitting [b(i),...,b(j)];
if c(k) = 3-r, then 3 divides the 0-splitting [b(j),...,b(k)];
if c(i) = c(k) = r, then 3 divides the 0-splitting [b(i),...,b(j),...,b(k)].
Therefore, inequality y <=2 must hold, and n <= 2*y <= 2*2 = 4.

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Emmanuel wrote on Set 15, 2019:

There is no  a(5).

It is even impossible to find numbers  m(1), m(2), m(3), m(4), m(5), m(6) that satisfies the following fifteen conditions :
m(i)&0&m(i+1) != 0 (mod 3)  for  i = 1, 2, 3, 4, 5
m(i)&0&m(i+1)&0&m(i+2) != 0 (mod 3)  for  i = 1, 2, 3, 4
m(i)&0&m(i+1)&0&m(i+2)&0&m(i+3) != 0 (mod 3)  for  i = 1, 2, 3
m(i)&0&m(i+1)&0&m(i+2)&0&m(i+3)&0&m(i+4) != 0 (mod 3)  for  i = 1, 2
m(1)&0&m(2)&0&m(3)&0&m(4)&0&m(5)&0&m(6) != 0 (mod 3)
(here, a&b  stands for the concatenation of  a  and  b).

The simplest way to prove this is to consider every of the 15 conditions modulo 3, using the fact that  a&b == a + b (mod 3).
Of course, this may be a bit tedious job to do 'by hand' : m(i) mod 3  has only three possible values whence there are  729  possible configurations to check.
I myself preferred to use a simple computer program.

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