The value of the smallest positive base ten integer that cannot be changed into a prime by changing a single digit was determined in Never prime!.

Determine the respective minimum values of a positive base N integer P that cannot be changed into a prime by changing a single digit, whenever N is a positive integer with 3 ≤ N ≤ 16, but N ≠ 10.

Note: P cannot contain any leading zero, and the first digit of P (reading left to right) cannot be changed to a zero.

using mathematica I found the required smallest numbers, interesting result is that all of them are 3 digit numbers in the respective base.  No suprise that they are not 1 or 2 digit numbers as that is impossible, but interesting that at least for values 3<=n<=50 there are none that require more than 3 digits.  Another interesting pattern is that the last digit appears to always be zero.  It would be nice to see a proof that these properties always hold true

below is my mathematica code used and its results

For[n=3,n„T50,++n,

num=2;

found=False;

While[!found,

dgs=IntegerDigits[num,n];

lng=Length[dgs];

pass=True;

If[!PrimeQ[num],

For[dp=1,dp„Tlng,++dp,

digit=dgs[[dp]];

For[d=0,d„Tn-1,++d,

If[d„jdigit,

If[!(dƒú0 && dpƒú1),

dgs2=dgs;

dgs2[[dp]]=d;

v=FromDigits[dgs2,n];

If[PrimeQ[v],

pass=False;

];

];

];

];

];

If[pass,

Print[n,": ",num," ",IntegerDigits[num,n]];

found=True;

];

];

++num;

];

];

3 : 24 {2,2,0}

4 : 24 {1,2,0}

5 : 90 {3,3,0}

6 : 90 {2,3,0}

7 : 119 {2,3,0}

8 : 200 {3,1,0}

9 : 117 {1,4,0}

10 : 200 {2,0,0}

11 : 319 {2,7,0}

12 : 528 {3,8,0}

13 : 1131 {6,9,0}

14 : 1134 {5,11,0}

15 : 525 {2,5,0}

16 : 1328 {5,3,0}

17 : 1343 {4,11,0}

18 : 1332 {4,2,0}

19 : 1330 {3,13,0}

20 : 1340 {3,7,0}

21 : 2478 {5,13,0}

22 : 7260 {15,0,0}

23 : 1334 {2,12,0}

24 : 5352 {9,7,0}

25 : 4300 {6,22,0}

26 : 5954 {8,21,0}

27 : 4833 {6,17,0}

28 : 13188 {16,23,0}

29 : 8468 {10,2,0}

30 : 10800 {12,0,0}

31 : 15686 {16,10,0}

32 : 11744 {11,15,0}

33 : 19338 {17,25,0}

34 : 19618 {16,33,0}

35 : 22575 {18,15,0}

36 : 19620 {15,5,0}

37 : 15688 {11,17,0}

38 : 28234 {19,21,0}

39 : 19617 {12,35,0}

40 : 25480 {15,37,0}

41 : 31406 {18,28,0}

42 : 19614 {11,5,0}

43 : 40291 {21,34,0}

44 : 25476 {13,7,0}

45 : 31410 {15,23,0}

46 : 31418 {14,39,0}

47 : 25474 {11,25,0}

48 : 69264 {30,3,0}

49 : 31409 {13,4,0}

50 : 31400 {12,28,0}

Posted by Daniel on 2010-09-04 19:22:49