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Primes with consecutive digits that differ exactly by 8.
+10
12
2, 3, 5, 7, 19, 191, 919, 919191919, 91919191919, 91919191919191919, 91919191919191919191919, 191919191919191919191919191919191
EXAMPLE
2 is a term since all its consecutive digits differ by 5 (there aren't any).
19 is a term because 1 and 9 differ by 8.
23 is not a term because its consecutive digits differ only by 1.
MATHEMATICA
Module[{nn=500, nine, one}, one=Select[Table[FromDigits[PadRight[{}, n, {1, 9}]], {n, nn}], PrimeQ]; nine=Select[Table[FromDigits[PadRight[{}, n, {9, 1}]], {n, nn}], PrimeQ]; Sort[Join[{2, 3, 5, 7}, nine, one]]] (* Harvey P. Dale, Jan 04 2023 *)
Primes with consecutive digits that differ exactly by 2.
+10
10
2, 3, 5, 7, 13, 31, 53, 79, 97, 131, 313, 353, 757, 797, 31357, 35353, 35753, 35797, 75353, 75797, 79757, 97579, 131357, 135353, 135757, 353531, 531353, 535757, 575753, 579757, 757579, 797579, 975313, 975797, 979757, 1313579, 3131353
MATHEMATICA
Join[{2, 3, 5, 7}, Select[Prime[Range[230000]], Union[Abs[ Differences[ IntegerDigits[ #]]]]=={2}&]] (* Harvey P. Dale, Nov 03 2013 *)
Primes with consecutive digits that differ exactly by 3.
+10
8
2, 3, 5, 7, 41, 47, 14741, 14747, 74747, 1414741, 1474141, 7414741, 4141414747, 4147414147, 14141414141, 14141414741, 14141474741, 14141474747, 14147414741, 14147474141, 14147474147, 14741414747, 74141414147, 74141414741, 74147414741, 74741474741, 74747414141
COMMENTS
All terms with more than a single digit must comprise only the digits 1, 4, and 7, because no number comprising the digits 2, 5, and 8 or the digits 3, 6, and 9 can be prime. - Harvey P. Dale, Mar 01 2023
MATHEMATICA
Join[{2, 3, 5, 7}, Table[Select[FromDigits/@Tuples[{1, 4, 7}, n], PrimeQ[#]&& Union[ Abs[ Differences[ IntegerDigits[ #]]]]=={3}&], {n, 11}]//Flatten] (* Harvey P. Dale, Mar 01 2023 *)
Primes with consecutive digits that differ exactly by 5.
+10
7
2, 3, 5, 7, 61, 83, 383, 727, 72727, 94949, 1616161, 383838383, 727272727, 383838383838383, 38383838383838383, 72727272727272727, 94949494949494949, 383838383838383838383
MATHEMATICA
Module[{nn=50, w1, w2}, w1=Flatten[Table[Select[FromDigits/@Table[ PadRight[ {}, n, {a, a+5}], {n, 2, nn}], PrimeQ], {a, 4}]]; w2=Flatten[Table[Select[ FromDigits/@ Table[PadRight[{}, n, {a+5, a}], {n, 2, nn}], PrimeQ], {a, 4}]]; Join[ {2, 3, 5, 7}, w1, w2]//Union] (* Harvey P. Dale, Jan 09 2021 *)
Primes with consecutive digits that differ exactly by 7.
+10
7
2, 3, 5, 7, 29, 181, 929, 18181, 929292929, 18181818181818181818181818181818181818181818181818181818181818181818181818181
MATHEMATICA
Module[{s18, s81, s29, s92}, s18=Select[Table[FromDigits[PadRight[{}, n, {1, 8}]], {n, 1, 181, 2}], PrimeQ]; s81=Select[Table[FromDigits[PadRight[{}, n, {8, 1}]], {n, 2, 182, 2}], PrimeQ]; s29 = Select[ Table[FromDigits[PadRight[{}, n, {2, 9}]], {n, 2, 182, 2}], PrimeQ]; s92 =Select[Table[ FromDigits[ PadRight[{}, n, {9, 2}]], {n, 1, 183, 2}], PrimeQ]; Join[{2, 3, 5, 7}, s18, s81, s29, s92]//Sort] (* Harvey P. Dale, Mar 23 2023 *)
Primes with consecutive digits that differ exactly by 6.
+10
6
2, 3, 5, 7, 17, 71, 1717171717171717171717171717171, 1717171717171717171717171717171717171
COMMENTS
Terms with more than 1 digit have digits alternating between 1 and 7.
No more terms < 10^3000. (End)
PROG
(PARI) upto(limit)={my(L=List([t|t<-[2, 3, 5], t<=limit]), m=1); while(m<limit, foreach([m*17\10, m*71\10, m*17, m*71], t, if(isprime(t)&&t<=limit, listput(L, t))); m=m*100+1); Vec(L)}
Prime worms [successive digit differences with absolute value of 4].
+10
3
151, 373, 95959, 9515959, 159595151, 159595951, 15151595951, 15951595151, 95951515159, 1515159515951, 1515959515951, 1515959595151, 1595159515151, 1595159595151, 9515151515159, 9515159515159, 9515159595959, 9595159515959
REFERENCES
Carlos Rivera's primepuzzles.net, Puzzle 246
FORMULA
Select prime numbers having the same first and last digits; if the uniform absolute value of successive digit differences is 4, add to sequence.
EXAMPLE
a(1)=373; first and last digits are 3; abs(3-7)=4; abs(7-3)=4; the worm is 3.
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