A224164 Prime numbers with m^2 digits that, if arranged in an m X m matrix, form m-digit reversible primes in each row and column and along the main diagonal.

2, 3, 5, 7, 1117, 1171, 7331, 131983991, 179907191, 179991179, 191199311, 191739971, 191797919, 199199311, 199709971, 199937971, 337353739, 373151113, 733353337, 797389337, 919311739, 971727179, 1193100992213191, 1193120192093911, 1193123793413719

Offset: 1

Author

Martin Renner, Mar 31 2013

Keywords

• nonn
• base

Comments

For m = 1, 2, 3, 4:

- number of terms with m^2 digits: 4, 3, 15, 86.

- smallest term with m^2 digits: 2, 1117, 131983991, 1193100992213191.

- largest term with m^2 digits: 7, 7331, 971727179, 9931722992931193.

Palindromic terms: 2, 3, 5, 7, 733353337, 971727179, ...

There are 1303816 terms with 25 digits, from 1119710007309831033317939 to 9979399989793939049937997, while the terms with 36 digits range from 111119100049100049150607134777979313 to 999931999983999983792293733331319919. - Giovanni Resta, Apr 05 2013

Examples

			a(5) = 1117 is the smallest 4-digit prime that if arranged in a 2 X 2 matrix yields in each row and column and along the main diagonal a number that is prime in both directions, i.e.,
  11
  17
-> 11 (4 times), 17 (3 times), 71 (3 times) are all reversible primes.
a(8) = 131983991 is the smallest 9-digit prime that if arranged in a 3 X 3 matrix yields in each row and column and along the main diagonal a number that is prime in both directions, i.e.,
  131
  983
  991
-> 131 (4 times), 181, 199, 389, 983, 991 (each 2 times) are all reversible primes.
a(23) = 1193100992213191 is the smallest 16-digit prime that if arranged in a 4 X 4 matrix yields in each row, and column and along the main diagonal a prime in both directions, i.e.,
  1193
  1009
  9221
  3191
-> 1009, 1021 (2 times), 1193 (3 times), 1201 (2 times), 1229, 1913, 3191, 3911 (3 times), 9001, 9029, 9209, 9221 are all reversible primes.
		

References

• Chris K. Caldwell, G. L. Honaker, Jr.: Prime Curios! The Dictionary of Prime Number Trivia. CreateSpace 2009, p. 219, and 229.

Links

• Martin Renner and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 108 terms from Martin Renner)
• Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 733353337.
• Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 3391382115599173.
• Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 19973...37991 (25-digits).

Crossrefs

Cf. A224398.

Programs

Maple

# Maple program generating all 4-digit primes
M:={}: for a in [1,3,7,9] do for b in [1,3,7,9] do if isprime(10*a+b) and isprime(10*b+a) then for c in [1,3,7,9] do for d in [1,3,7,9] do if isprime(10*c+d) and isprime(10*d+c) and isprime(10*a+c) and isprime(10*c+a) and isprime(10*b+d) and isprime(10*d+b) and isprime(10*a+d) and isprime(10*d+a) then S:=[a,b,c,d]: if isprime(add(S[j]*10^(4-j),j=1..4)) then M:={op(M),add(S[j]*10^(4-j),j=1..4)}: fi: fi: od: od: fi: od: od: M;
# Maple program generating all 9-digit primes
M:={}: for d in [1,3,7,9] do for e from 0 to 9 do for f in [1,3,7,9] do if isprime(100*d+10*e+f) and isprime(100*f+10*e+d) then for a in [1,3,7,9] do for b in [1,3,7,9] do for c in [1,3,7,9] do if isprime(100*a+10*b+c) and isprime(100*c+10*b+a) then for g in [1,3,7,9] do for h in [1,3,7,9] do for i in [1,3,7,9] do if isprime(100*g+10*h+i) and isprime(100*i+10*h+g) and isprime(100*a+10*d+g) and isprime(100*g+10*d+a) and isprime(100*b+10*e+h) and isprime(100*h+10*e+b) and isprime(100*c+10*f+i) and isprime(100*i+10*f+c) and isprime(100*a+10*e+i) and isprime(100*i+10*e+a) then S:=[a,b,c,d,e,f,g,h,i]: if isprime(add(S[j]*10^(9-j),j=1..9)) then M:={op(M),add(S[j]*10^(9-j),j=1..9)}: fi: fi: od: od: od: fi: od: od: od: fi: od: od: od: M;