Problems & Puzzles: Puzzles

Puzzle 651 6866683 Look at this set of primes: +6866683 6+866683 68+66683 686+6683 6866+683 68666+83 686668+3 Q. Send your largest example (no zeros allowed anyplace)

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Contributions came from W. Edwin Clark, Emmanuel Vantieghem, J. K. Andersen & Rayn Bailey.

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Clark wrote:

I found no larger example. The only primes < 10^15 that satisfy the condition
 are these 45 primes:

 

2,3,5,7,11,23,29,41,43,47,61,67,83,89,227, 229,281,443,449,467,647,
661,683,821,863,881,2221,2267,2281,2447,4229,4463,4643,8221,8821,
26881,28429,68683,282881,282889,464447,626261,864883,6462667,6866683

Is it possible that this sequence is finite?  I cannot find it in the OEIS.  However, if one allows 0 as a digit

in the prime then one obtains the sequence of "magnanimous primes" https://oeis.org/A089392

 

According to Gerry Myerson's remark at the bottom of the page

 http://math.stackexchange.com/questions/162042/infinitely-many-prime-numbers-with-prescribed-digits

it is even unknown whether or not there are infinitely many primes that do not contain 0 in their decimal representation.
 

-------generalizations-------------------->

 

If we do this problem for an arbitrary base b then a prime p satisfies the condition

if and only if  for each i from 1 to floor(log[b](p)) the numbers rem(p,b^i) + quo(p,b^i) are prime,

where rem(p,m) ,(resp, quo(p,m)) is the remainder,(resp. quotient) when p is divided by m.

 

Let GoodPrimes[b] be the set of such primes and let GoodPrimesNo0[p] be the set of such primes

that have no 0 digit in their base b representation. Then I get the following data for the first

12 million primes:
 

|GoodPrimes[ 3]| =  12, largest = [1, 0, 1, 0, 0, 1, 1, 1, 1, 1] (in base 3)

|GoodPrimes[ 4]| =  15, largest = [2, 0, 0, 0, 2, 0, 2, 2, 3] (in base 4)

|GoodPrimes[ 5]| =  90, largest = [1, 0, 1, 2, 3, 0, 2, 2, 0, 3, 1] (in base 5)

|GoodPrimes[ 6]| =  18, largest = [2, 0, 0, 0, 0, 2, 5] (in base 6)

|GoodPrimes[ 7]| = 584, largest = [2, 3, 3, 6, 5, 5, 5, 0, 6, 6] (in base 7)

|GoodPrimes[ 8]| =  29, largest = [6, 0, 0, 6, 0, 6, 0, 5] (in base 8)

|GoodPrimes[ 9]| = 342, largest = [1, 6, 0, 4, 4, 1, 3, 7, 1] (in base 9)

|GoodPrimes[10]| =  80, largest = [4, 8, 8, 0, 4, 8, 0, 9] (in base 10)

 

 

|GoodPrimesNo0[ 3]| =   4, largest = [1, 1, 1] (in base 3)

|GoodPrimesNo0[ 4]| =   8, largest = [2, 2, 2, 2, 2, 1] (in base 4)

|GoodPrimesNo0[ 5]| =  37, largest = [4, 1, 4, 1, 1, 4, 1, 3, 3, 3] (in base 5)

|GoodPrimesNo0[ 6]| =   8, largest = [4, 4, 4, 1] (in base 6)

|GoodPrimesNo0[ 7]| = 290, largest = [5, 2, 2, 5, 2, 2, 4, 5, 3, 1] (in base 7)

|GoodPrimesNo0[ 8]| =  17, largest = [6, 6, 6, 6, 1] (in base 8)

|GoodPrimesNo0[ 9]| = 187, largest = [3, 3, 8, 3, 2, 3, 5, 5, 5] (in base 9)

|GoodPrimesNo0[10]| =  45, largest = [6, 8, 6, 6, 6, 8, 3] (in base 10)

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Emmanuel wrote:

If there is a bigger solution than  6866683, it must have 20 digits or more.

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Andersen wrote:

A search found no larger example up to 19 digits. According to
heuristics, 6866683 is probably the largest there is.
It is listed in http://oeis.org/A103548
[Note by CR: this sequence does not match with the complete list given above by Mr. Clark]

All digits except the last must be even. Otherwise one of the
sums would be even. For each possible ending digit there is one even digit which cannot be in the number due to divisibility by 5. For 1, 3, 7, 9
it is respectively 4, 2, 8, 6. That leaves relatively few possibilities to test in a search.

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Ryan wrote:

 I checked many numbers up to 8848822224848863, and have found no solutions.

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