Problems & Puzzles: Puzzles

 Puzzle 776. Ten consecutive integers such that... José de Jesús Camacho sent the following nice puzzle:   Using the transformation N -> M such that M = N + Σ (di^2), where di are the digits of N, we are looking only for N integers such that M is a prime number.   Example:   N= 10 then M =  10 + 1^2 + 0^2 = 11   Moreover we are looking for sets of consecutive integers N such that all the corresponding M values are prime integers.   One set having 10 members is {N}= {10, 11, ... 19}, because the corresponding M values are al prime numbers {M} = { 11, 13, 17, 23, 31, 41, 53, 67, 83, 101}   Carlos Rivera have discovered another set {N}, having 10 consecutive integers and 10 prime numbers in the set {M}, being the first N value 1761702690   Q.  Find 3 more values as 10 and 1761702690

Contributions came from Giovanni Resta, Flavio Torasso, Jan van Delden.

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Giovanni wrote:

I found all the solutions up to 10^15, then I deleted the result by accident, then I recomputed all the solutions up to 2*10^14 (after that I needed my PC for other things).

There are 711 such numbers up to 2*10^14, the first 30 being
10, 1761702690, 7226006660, 16453361570, 95748657190, 104217487100, 111058349320, 141665059420, 168759510430, 177313689280, 177313689330, 178209124090, 188343072120, 347296044930, 347296045010, 381449093790, 381449093840, 445151776780, 491570264380, 502344967120, 610260789170, 629827603880, 658654806500, 692148762400, 769651428120, 885255835150, 968235529460, 972397702210, 1144051661370, 1493824020460, and the
last 5 are
196671162186470, 197753259684960, 198207055214690, 198208971208570, 198440000999040.

Some numbers lead to the same 10 primes, for example,
177313689280 and 177313689330 or 347296044930 and 347296045010.

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Flavio wrote:

I found these three values satisfying Puzzle 776:

7226006660
16453361570
95748657190

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Jan wrote:

If the sequence of N[i] doesn’t have a carry (hence maximal length 10, the last digit of N[i]=i) the corresponding differences between M[i] and M[1] depend only on the last digit and form an admissible constellation with M[i]-M[1] in [0,2,6,12,20,30,42,56,72,90]. This means that the values for M[1] must be  either {11,17,41,101,137,167} mod (2*3*5*7), which restricts the search.

If there is a carry the digits of N[1] before the last digit must be of the form {9}[2k+1]. This can be seen by using a parity check on the corresponding M[i]. For each of these forms one can compute the maximal sequence of N by checking whether there are still some free residues for M[1] left.

The result is that if N[1] ends with the number i (and we have a carry), the maximal length is 11-i.

90     the maximal length is 11 with 24 free residues mod (2*3*5*7*11)
9990 the maximal length is 11 with 12 free residues mod (2*3*5*7*11)
The free residues for {9}[k]0 seem to be a subset of {9}[5]1.

91     the maximum length is 10 with 6 free residues mod (2*3*5*7)
9991 the maximum length is 10 with 3 free residues mod (2*3*5*7)
The free residues for {9}[k]1 seem to be a subset of {9}[1]0.

etc.

I checked M[1]<39816*10^6, only when no carry is present, length 10, and found:
N[1]                M[1]  residue
10                    11    11
1761702690    1761702947  137
7226006660    7226006861  101
16453361570  16453361777  167

I checked length 11, only where N ends in 90, no solutions with M[1]<42966*10^6.

A few remarks.

This last routine was largely checking the same numbers, because the admissible constellation for length 11 is actually the same as given, with –70 on the last position. So all I needed to check was that for N[1]=1761702690 M[11]=1761702889 is not prime.

Multiple values for M[1] exist for which M[2]..M[11] are prime:
M[1]=25471289141 N[1]=25471288759 does not end in 90
M[1]=23953276661 N[1] does not exist
So they don’t generate solutions.

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