Problems & Puzzles: Puzzles

Puzzle 238. The hidden prime

This last two weeks our family started trying to solve some puzzles not of numerical nature but of the so called 'lateral thinking' type (in order to get used to think 'out of the box').

One of these puzzles (by Lloyd King) suggested to me a parallel prime puzzle. I hope none of you get angry with me for posting this, maybe very simple, puzzle:

12429793
124097
14239807
135697
7561
15209167
??????

Question: Find the
six digits missing number in the above sequence.

Solution:

At last one solution for a puzzle posted 15 years ago (2003), comes from James Lavelle on July 1, 2018:

Solution: 702991.

The sequence is a cycle of emirps with the middle two digits of each term the same as the second and next-to-last digits of the subsequent term.

Discussion:

I came across this puzzle on the OEIS as sequence A187399. As Arkadiusz Wesolowski observed, the first six terms are each an emirp — a prime that when written in reverse is a different prime. This is a helpful but insufficient to uniquely continue the sequence, as there are thousands of six-digit emirps.

Visual inspection of the terms of the sequence shows multiple repeated digits from one term to the next. One such repetition pattern is present in each sequential pair—the middle two digits become the second and next-to-last digits in the next term. For example, the middle two digits of the first term 12429793 are 2 and 9. Given the second term is a six digits, it must take the form ?2??9? to continue the pattern.

Based on this condition, the seventh term, which is also given to be a six digit number, must take the form ?0??9?. However, there are hundreds of primes and 99 emirps that meet this condition.

If we further assume this to be a cyclic sequence, the middle two digits would have to be 29 to fit with the first given term. This suggests that the seventh term must take the form ?0299?.

There are only five primes of this form: 202999, 302999, 402991, 602999 and 702991. Only one of these primes is also an emirp: 702991.

Other Observations:

While this is a unique solution for the six given terms, this is not the only cyclic sequence of emirps with these properties. For instance, the second term can be replaced with 924097 and meet the conditions described above.

There does not appear to be a consistent relationship between the terms governing the length of each term. However, given the first six terms, there are no emirps shorter than six digits that meet the above conditions for the seventh term. Six digits is the shortest emirp length that uniquely continues the sequence.

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