Problems & Puzzles:
Collection 20th
Coll.20th-001.
A
Puzzle-sequence
On
March 10, 2018, G. L. Honaker posed the following puzzle sequence in the
middle of a mailing list:
Puzzle
Sequence: 7, 19, 79, 491, 2423, ... . Find the next term.
BTW, this puzzle
sequence was already published in his site
here.
The puzzle was solved
by Fausto Morales eleven days later, on March 21, 2018: The next term is
8147. Here is his explanation:
“Having
observed that, for n=1 to 5, the nth term has always been the lowest prime
such that the 1st n consecutive gaps between consecutive primes following it
are precisely each of the first n odd primes plus 1, my proposed solution is
8147, because the next 6 gaps above it are as follows:
8161 - 8147 = 14 = 13 + 1
8167 - 8161 = 6 = 5 + 1
8171 - 8167 = 4 = 3 + 1
8179 - 8171 = 8 = 7 + 1
8191 - 8179 = 12 = 11 + 1
8209 - 8191 = 18 = 17 + 1”
G. L. Honaker has accepted the Mr. Morales solution.
Q. So, the only remaining question is this: find the next term if it exists
or prove that it does not exist.
Contributions came from Emmanuel Vantieghem, Jan van Delden
and Giovanni Resta.
***
Emmanuel wrote:
Suppose there is a next
term p.
Then p is the first
member of a set P = {p, p1, p2, p3, p4, p5, p6, p7}
of eigth consecutive
primes with successive gaps {g1, g2, g3, g4, g5, g6, g7}
which are a permutation
of {4, 6, 8, 12, 14, 18, 20}.
I examined all possible
permutations and these was my observation :
whatever p is mod 3
(1 or 2), there is allways some i <= 7 such that p+g1+ ... +gi (=
pi) is divisible by 3.
So, there is no next
term !...
Maybe I have been
too quick (which is never OK).
It could be
possible that the 'next' term is a prime p such that the ten
successive primes
have gaps a
permutation of {4, 6, 8, 12, 14, 18, 20, 24, 30, 32}.
I'll give it a try,
but I'm afraid i will run out of time ...
***
Jan wrote:
There is no next term.
The consecutive gaps g[i] must be in the set
{4,6,8,12,14,18,20}.
Consider a permutation of this set: [g[i]] and a starting
prime p[1]. The primes are now defined by p[i+1]-p[i]=g[i].
So one could write p[i+1]=p[1]+sum(g[j],j=1..i). Or in words, each prime
is now the cumulative sum of the gaps in the permutation+ a starting
prime. This list of cumulative sums must be admissible mod 3, mod 5 and
mod 7.
If p[1]<=347 then the set of consecutive gaps contains the number 2.
If one wants to find a solution, we must have a residue set mod p
(p=(2),3,5,7) such that:
-
The number of residues
is smaller than p [We must be able to choose a p[1] such that
p[i+1]<>0 mod p for all i] (I)
- If there are
p-1 residues each unequal to 0 mod p, than p[1] must be 0 mod p. But
p=3,5,7 is not possible.
So if 0 is not present, the number of free residues must be smaller
than p-1 in order to be able to choose a starting prime p[1].(II)
The permutation [4,6,8,12,14,18,20] defines gaps relative
to p[1]: [4,10,18,30,44,62,82].
Modulo 3 it reads: [1,1,0,0,2,2,1], so the residue set
mod 3={0,1,2}, which is a full residue set. (I)
The permutation [8,6,12,18,14,4,20] defines gaps
relative to p[1]: [8,14,26,44,58,62,82].
Modulo 3 it reads: [2,2,2,2,1,2,1], so the residue set mod 3={1,2}, p
must be 0 mod 3, but p=3 is not a solution.
A check reveals that there are no permutations that
comply to rule (I) and rule (II) modulo 3.
It is possible to give a proof, without computing all permuations, which
uses the fact that 82 is always present, which is 1 mod 3.
***
Giovanni wrote:
The next term does not exist.
To have a next term, the gaps between the primes must be
equal to a permutation of 1 + the first 7 odd primes
1 + {3, 5, 7, 11, 13, 17, 19} = {4, 6, 8, 12, 14, 18,
20}.
If the first prime (the solution) is p, the following
primes are determined by the gaps, like p+4, p+4+6,
p+4+6+8,
etc. if the gaps are sorted as above.
These gaps, modulo 3, are equal to {1, 0, 2, 0, 2, 0, 2}.
It is easy to see that the partial sums of these gaps,
whatever is their permutation,
always contain at least a 1 and at least a 2 modulo 3.
For the case above, the sums are equal to
{1, 1+0, 1+0+2, 1+0+2+0, ...} =
{1,1,3,3,5,5,7} == {1,1,0,0,2,2}.
If the gaps are in another order, say {0,2,2,1,0,0,2},
the resulting sums are {0,2,4,5,5,5,7} = {0,2,1,2,2,2,1}.
Now, the first prime of the sequence (the solution we
are looking for) is greater than 3 and thus it is
either equal to 1 or equal to 2 mod 3.
But this means that building the other 7 primes starting
from
p and using the gaps always result in a number which is
1+2==0
or 2+1 ==0 mod 3, thus a multiple of 3 that cannot be a
prime.
Hence there cannot be a next term.
***
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