Problems & Puzzles: Puzzles

 Puzzle 377. Add 2 to any digit Recently I saw this curio: Add 2 to any digit of 32057 [one at a time] and it will remain prime. [Blanchette] Question. Can you find a larger example?

Contributions came from Jaroslaw Wroblewski, Farideh Firoozbakht, Jean-Charles Meyrignac, & Giovanni Resta

Jaroslaw & Farideh soon found that:

There is no larger example. The only 5-digit numbers are: 21011, 24371,
32057. For at least 6-digit number p you require in particular the following 7
numbers to be prime: p, p+2, p+20, p+200, p+2000, p+20000, p+200000,
but they give 7 distinct residues mod 7, hence one of them is divisible
by 7 and therefore not prime.

***

Then Farideh came up with the variation #1 for this puzzle (digits may be 8 or 9):

3, 5, 11, 17, 41, 89, 107, 137, 197, 347, 419, 809, 839, 1229, 1259, 1559,
1931,2111, 2591, 2657, 3527, 4421, 5639, 5669, 6761, 7109, 8069, 21011,
24371, 30839, 32057, 37139, 78887, 97787, 181439, 735719, 11879249,
24504917, 31310891, 79348481,... are  such numbers.

79348481 is in the sequence because all the nine numbers 79348481, 99348481, 711348481, 79548481, 79368481, 793410481, 79348681, 793484101 & 79348483  are primes.

Note that 32057 is the largest number n such that n has the requested property and all digits of n are less than 8.

458481407 is a larger solution. Namely all the ten numbers, 458481407, 658481407, 478481407, 4510481407, 458681407, 4584101407, 458483407, 458481607, 458481427 & 458481409 are primes.

Is 458481407 the largest solution for the variation #1?

***

Jean-Charles got another interesting variation #2:

...if we accept that the base value is not a prime, there seems to be an infinite number of solutions ! For example:150406347, 211455657, 527067401, 530402747... (I stopped my program just after the last solution)... Up to 10^11, there is only another solution: 7101620477...here is a bigger one: 21072701741...A bigger one: 53175232341...At this moment, I tested all values below 132000000001

In total he got 104 solutions, 16 if the base is a prime number (the largest is 32057), 88 if the base is composite (the largest is 53175232341).

***

Giovanni Resta independently worked over the Meyrignac's variation #2 and its symmetric: 'the base value is not prime and subtract 2', variation #3

For the variation #2, he obtained exactly the same values than JCM except one more & larger solution, 13 digits large: 7134354660741

For the composite base subtracting 2, variation #3, his largest solution was: 92997492273.

***

Can you get a larger solution than 7134354660741 & 92997492273 , for the variations #2 & #3, respectively?
Are there finite or infinite solutions for the variations #1, #2 & #3?

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