Problems & Puzzles: Puzzles

Puzzle 337. p inside p2. I have gotten the following results: p prime is inside p2: 5       25 66952741        4482669527413081 146509717       21465097175420089 753348181       567533481816008761 If we accept to find p in p2, in reverse order: 5       25 87690769        7689670967811361        87690769 311314139       96916493141311321       311314139   Question: 1. Can you find a few more (next) cases of each type? 2. Do you devise how to get large examples?

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Contributions came from Faride Firoozbakht, Giovanni Resta & Dan Dima.

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Faride & Dan found none example in their own search up to 12660897863 and 4,280,000,000, respectively.

Giovanni found three more examples of the first type and one of the second one up to 3,266,358,692,473 and 1,972,348,725,227, respectively:

562984507451^2 = 31695155(562984507451)7401
860628177919^2 = 740680(860628177919)170561
978058181203^2 = 9565(978058181203)82527209

and

44466725441^2 = 197728967(144527664448)1

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Faride & Giovanni devised the same alternative solutions to question 2:

"if p is of the form 5*10^n-1 then p appears at the end of p^3"

Faride added " 5*10^29282-1 is the largest known probable prime of the form 5*10^n-1(please see sequences A056712 & A093945)". Giovanni gave no specific example.

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J. K. Andersen added:

Regarding what Faride added, The Prime Pages at http://primes.utm.edu says: 5*10^29282-1 is a proven prime, found by Eric J. Sorensen with Yves Gallot's Proth.exe in 2001. Sorensen also found and proved 5*10^78790-1 in 2005. The form with p+1 completely factored makes them easily provable

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