Problems & Puzzles: Puzzles

Puzzle 402. 1263560563 Perhaps you already know the following curio by Blanchette: 58n + 38n + 05n + 23n + 33n is prime for n = 0 to 10. Note that 5838052333 is a 10-digit prime. [Blanchette] Unexpectedly, I found two more but smaller examples of this type: 1263560563 & 5605126363 (which is a recombination of the first one), both primes for n=0 to 10. Extending the idea, the smallest example that I have found of primes abcd... such that a^i+b^i+... is prime for n=0 to 9 is 1113997. Questions. Q1. Find an example of the Blanchette type prime for n=0 to 11 or more. Q2. Find an example of my type prime for n=0 to 10 or more.

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Contributions came from Adam Stinchcombe, Giovanni Resta, Luke Pebody, Farideh Firoozbakht & Fred Schneider.

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All of them noticed that prime repunits are infinite trivial solutions to Q2.

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Resta wrote:

For question Q1 I've searched all the numbers with 2x7=14 digits.
The smallest prime which beats n=10 reaches directly n=12 and it is
01 03 33 07 58 90 47 if leading zero is allowed.
If leading zero it is not allowed then we have
n=11 for 10 03 14 12 14 23 07
n=12 for 16 02 07 24 23 28 27

The next possible length is 2x11=22 and the search space is too large for
me. The best I found is n=13, attained by
01 01 01 08 24 24 52 79 78 74 77 (leading zero) or
24 01 01 01 08 24 52 74 77 78 79,
but probably these primes are neither minimal nor optimal.

For question Q2, I've extended a little my previous search,
for the two cases 0^0=undefined, (so only zeroless primes allowed)
and 0^0=1.

For both cases, I think I've found the smallest prime which satisfies
the relation for 0..k where k ranges from 2 to 15. The values are reported in the attached table. Since the numbers are quite long (up to 61 digits for the case
n=15, zeroless primes) I've used the convention that D_k means digits D repeated k times.

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Pebody wrote:

Your type for n=0..10:2,022,226,866,629 (Smallest)

Smallest for n=0..11 is also smallest for n=0..12: 10012225556799799

Smallest for n=0..13 is 10000122233333334445888588583

Smallest for n=0..14 is 1011111111133333333344455666543

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Farideh wrote:

Answer to Q2: It's obvious that all repunit primes are primes of the second type for all n's.

 

Also all 4507-digit primes of the form (10^4507-1)/9+3(10^m+10^r+10^s+10^t) where 0<m<r<s<t<4507 are primes of this type for n = 0,1,..., 12 & 13. ".

Because such primes have 4503 ones & four 4's and all 14 numbers 4503+ 4*4^k (-1<k<14)  are prime.

 

p = 1(1198).4.1(3304).4441 = (10^4507-1)/9+(10^1+10^2+10^3+10^3308)  &

q = 1(3577).4.1(923).41441 = (10^4507-1)/9+3(10^2+10^3+10^5+10^929) are two of them.

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Fred wrote:

Q1) 1033307589047 is the minimal Blanchette prime for n=0 to 11 (in
fact, it holds for n=12 as well)

Q2) Note that no digit can be zero because 0^0 is undefined.

With these points in mind, I found the following minimal non-trivial solutions:

n minimal prime such that the n is the highest power
10 11111111225256599 (17 digits)
11 12223333333334558899889 (23 digits)
12 1111122233444555999 (19 digits)
13 11333334555577777777777777777777777888799 (41 digits)
14 1133333344444444455555555666977777779 (31 digits)

There's no solutions s for n=15 such that s < 10^42
Note: That MANY other solutions for a given n can be found just by
randomly permutating the different digits for a given solution above.
In fact, my approach was find all the orderings of digits (or blocks
of digits) such that they were non-decreasing. Storing each block in
an array eliminated the need to do any division. Only adding was
necessary (once the prime powers were pre-computed).
Once a solution was found I could zero in on the minimum prime
solution usually by just permutating the last few digits of the
non-decreasing ordering. Inspecting the solutions should clarify what
I mean.

Bonus (for blocks of 3 digits):
When considering blocks of 3 digits (rather than blocks of 2 and 1
respectively in Q1 and Q2), I found the minimal number such that the
relation holds for n=0 through 11:

1,031,588,010,209

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