Problems & Puzzles: Puzzles

 Puzzle 488. Follow up to Puzzle 487 My friend Patrick Capelle sent the following extension to Puzzle 487 5*7*13*19 - 2*3*11*17 = 7523, written with the first consecutive primes on the left side of the equation. Q1. Is 7523 the largest such prime with only prime digits? Q2. Is 7523 the largest such left-truncatable prime? Q3. Is 7523 the largest such emirp?

Anton Vrba wrote (May 09)

Patrick’s example is unique insofar that the same prime satisfies all three questions. Considering each question individually I can offer following larger examples:

Q1: 3*7*19*23*29*31*41 - 2*5*11*13*17*37 = 337355573 (no digit 2)
and 7*11*13*31*37*41 + 2*3*5*17*19*23*29 = 53537257 (all four prime digits)

Q2: 2*3*13*17*19*23 – 5*7*11*29*31 = 233347
or 2*3*7*13*17*19*29 - 5*11*23*31 = 5075167 if digit zero is allowed.

Q3: 733#/(2*109*157) - 2*109*157 is a 300 digit emirp.

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