Hard code golf: Regex for divisibility by 7

Question

Matthias Goergens has a 63,993-character regex to match numbers divisible by 7, but that includes a lot of fluff: redundant parentheses, distribution (xx|xy|yx|yy rather than [xy]{2}) and other issues, though I'm sure a fresh start would be helpful in saving space. How small can this be made?

Any reasonable variety of regular expressions are allowed, but no executable code in the regex.

The regular expression should match all strings containing the representation of a decimal number divisible by 7 and no others. Extra credit for a regex that does not allow initial 0s.

Answer 1

13,755 12,701 Characters

0|(7|[18]5*4|(2|9|[18]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4)))|(5|[18]5*[29]|(2|9|[18]5*6)(3|[29]5*6)*(6|[29]5*[29])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(6|[07]5*4|(1|8|[07]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))))|(6|[18]5*3|(2|9|[18]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))|(5|[18]5*[29]|(2|9|[18]5*6)(3|[29]5*6)*(6|[29]5*[29])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(4|[18]5*[18]|(2|9|[18]5*6)(3|[29]5*6)*(5|[29]5*[18])|(3|[18]5*[07]|(2|9|[18]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(5|[07]5*3|(1|8|[07]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))))(2|9|45*3|(5|45*6)(3|[29]5*6)*(0|7|[29]5*3)|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))|(1|8|45*[29]|(5|45*6)(3|[29]5*6)*(6|[29]5*[29])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(5|[07]5*3|(1|8|[07]5*6)(3|[29]5*6)*(0|7|[29]5*3)|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(1|8|35*3|(4|35*6)(3|[29]5*6)*(0|7|[29]5*3)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(4|65*3|(0|7|65*6)(3|[29]5*6)*(0|7|[29]5*3)))))*(3|45*4|(5|45*6)(3|[29]5*6)*(1|8|[29]5*4)|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4)))|(1|8|45*[29]|(5|45*6)(3|[29]5*6)*(6|[29]5*[29])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(0|7|45*[18]|(5|45*6)(3|[29]5*6)*(5|[29]5*[18])|(6|45*[07]|(5|45*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))(4|[07]5*[29]|(1|8|[07]5*6)(3|[29]5*6)*(6|[29]5*[29])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(0|7|35*[29]|(4|35*6)(3|[29]5*6)*(6|[29]5*[29])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(3|65*[29]|(0|7|65*6)(3|[29]5*6)*(6|[29]5*[29]))))*(6|[07]5*4|(1|8|[07]5*6)(3|[29]5*6)*(1|8|[29]5*4)|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))|(3|[07]5*[18]|(1|8|[07]5*6)(3|[29]5*6)*(5|[29]5*[18])|(2|[07]5*[07]|(1|8|[07]5*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))(6|35*[18]|(4|35*6)(3|[29]5*6)*(5|[29]5*[18])|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(2|9|65*[18]|(0|7|65*6)(3|[29]5*6)*(5|[29]5*[18])))*(2|9|35*4|(4|35*6)(3|[29]5*6)*(1|8|[29]5*4)|(5|35*[07]|(4|35*6)(3|[29]5*6)*(4|[29]5*[07]))(1|8|65*[07]|(0|7|65*6)(3|[29]5*6)*(4|[29]5*[07]))*(5|65*4|(0|7|65*6)(3|[29]5*6)*(1|8|[29]5*4))))))*

This is tested with The Regex Coach

How We Get There

The Regex above produced by first constructing an NFA which would accept the input we want (decimals divisible by 7) and then converting to a Regular Expression and fixing the notaion

To understand this, it helps to first make an NFA which accepts the following language:

L = {w | w is a binary representation of an integer divisible by 7 }

That is, it will 'match' binary numbers that are divisible by 7.

The NFA looks like this:

How it works

You keep a current value A that represents the value of the bits the NFA has read. When you read a 0 then A = 2*A and when you read a 1 A = 2*A + 1. At each step you calculate A mod 7 then you go to the state that represents the answer.

So a test run:

We're reading in 10101 which is the binary representation for 21 in decimal.

1. We start at state q0, currently A=0
2. We read a 1, from the 'rule' above A = 2*A + 1 so A = 1. A mod 7 = 1 so we move to state q1
3. We read a 0, A = 2*A = 2, A mod 7 = 2 so we move to q2
4. Read a 1, A = 2*A + 1 = 5, A mod 7 = 5, move to q5
5. Read a 0, A = 2*A = 10, A mod 7 = 3, move to q3
6. Read a 1, A = 2*A + 1 = 21, A mod 7 = 0, move to q0
7. The input is accepted so the number 10101 is divisible by 7!

Converting the NFA to a Regular Expression is a tricky task so I got JFLAP to do it for me, producing the following:

0|(111|100((1|00)0)*011|(101|100((1|00)0)*(1|00)1)(1((1|00)0)*(1|00)1)*(01|1((1|00)0)*011)|(110|100((1|00)0)*010|(101|100((1|00)0)*(1|00)1)(1((1|00)0)*(1|00)1)*(00|1((1|00)0)*010))(1|0(1((1|00)0)*(1|00)1)*(00|1((1|00)0)*010))*0(1((1|00)0)*(1|00)1)*(01|1((1|00)0)*011))*

For Decimal Numbers

The process is much the same:

I constructed a NFA which accepts the language:

L = {w | w is a decimal number that is divisible by 7}

Here is the NFA:

The logic is similar, same number of states just many more transitions to handle all the extra digits decimal numbers bring.

Now the rule to change A at each step is: when you read a decimal digit n: A = 10*A + n. Then again you just mod A by 7 and go to the next state.

Another edit

The above regular expression now rejects numbers leading zeros - apart from zero itself of course

This makes the NFA slightly different, basically you branch off from the initial node when you read the first zero. Reading another zero puts you into an infinite loop on the branched state. I haven't fixed the diagram to show this.

Edit again

Did some "metaregex" and shortened my regex by replacing some of the unions with character classes.

Answer 2

468 characters

Ruby's regex flavor allows recursion (although it's sort of cheating), so it is straightforward to implement a DFA that recognizes numbers divisible by 7 using that. Each named group corresponds to a state, and each branch in the alternations consumes one digit and then jumps to the appropriate state. If the end of the number is reached, the regex matches only if the engine is in the "A" group, otherwise it fails.

It recognizes leading zeros.

(?!$)(?>(|(?<B>4\g<A>|5\g<B>|6\g<C>|[07]\g<D>|[18]\g<E>|[29]\g<F>|3\g<G>))(|(?<C>[18]\g<A>|[29]\g<B>|3\g<C>|4\g<D>|5\g<E>|6\g<F>|[07]\g<G>))(|(?<D>5\g<A>|6\g<B>|[07]\g<C>|[18]\g<D>|[29]\g<E>|3\g<F>|4\g<G>))(|(?<E>[29]\g<A>|3\g<B>|4\g<C>|5\g<D>|6\g<E>|[07]\g<F>|[18]\g<G>))(|(?<F>6\g<A>|[07]\g<B>|[18]\g<C>|[29]\g<D>|3\g<E>|4\g<F>|5\g<G>))(|(?<G>3\g<A>|4\g<B>|5\g<C>|6\g<D>|[07]\g<E>|[18]\g<F>|[29]\g<G>)))(?<A>$|[07]\g<A>|[18]\g<B>|[29]\g<C>|3\g<D>|4\g<E>|5\g<F>|6\g<G>)

Answer 3

I was really impressed by Griffin's answer and needed to figure out how it worked! The result is the following JavaScript. (It's 3.5k characters, which is shorter in a way!) The gen function takes a divisor and base and generates a regular expression that matches numbers in the specified base that are divisible by that divisor.

I've generalized Griffin's NFA for any base: the nfa function takes a divisor and base and returns a two dimensional array of transitions. The input required to go from state 0 to state 2, for example, is states[0][2] == "1".

The reduce function takes in the states array and runs it through this algorithm to translate the NFA to regex. The regexes that are generated are huge and look like they have a lot of redundant clauses, despite my attempts at optimization. The regex for 7 base 10 is about ~67k characters long; Firefox throws an "InternalError" for n > 5 trying to parse the regex; running the regex on Chrome starts getting slow for n > 6.

There's also the test function that takes a regex and base and runs it against the numbers 0 to 100, so test(gen(5)) == [0, 5, 10, 15, ...].

Despite the suboptimal result, this was a fantastic learning opportunity, and I hope some of this code will be useful in the future!

function gen(b, base) {
    var states = nfa(b, base)
    for (var i = 0; i < states.length; i++)
        states = reduce(states, i);
    return states[0][0] != 'phi' && new RegExp('^' + wrap(states[0][0]) + '$');
}

function test(reg, base) {
    if (!base)
        base = 10;

    var x = [];
    for (var i = 0; i < 100; i++)
        x.push(i);
    return x.map(function (a) {return a.toString(base)}).filter(reg.test.bind(reg)).map(function (a) {return parseInt(a, base)})
}

function nfa(b, base) {
    if (!base)
        base = 10;

    var states = [];
    for (var i = 0; i < b; i++) {
        states[i] = [];
        for (var j = 0; j < b; j++)
            states[i][j] = [];
    }

    for (var i = 0; i < b; i++)
        for (var n = 0; n < base; n++)
            states[i][(i * base + n) % b].push(n.toString());

    for (var i = 0; i < b; i++)
        for (var j = 0; j < b; j++)
            states[i][j] = states[i][j].length > 1 ? '[' + states[i][j].join('') + ']' : (states[i][j][0] || 'phi');
    return states;
}

// http://www.cs.umbc.edu/~squire/cs451_l7.html
function reduce(states, n) {
    var s = states.length;
    var reduced = [];
    for (var i = 0; i < s; i++) {
        reduced[i] = [];
        for (var j = 0; j < s; j++) {
            // reduced[i][j] = wrap(states[i][n] + wrap(states[n][n]) + '*' + states[n][j] + '|' + states[i][j]);
            reduced[i][j] = '';

            if (states[i][n] == 'phi' || states[n][j] == 'phi') {
                reduced[i][j] = states[i][j];
                continue;
            }

            if (states[i][n] != states[n][n])
                reduced[i][j] += wrap(states[i][n]);

            if (states[n][n] != 'phi') {
                reduced[i][j] += wrap(states[n][n]);

                if (states[i][n] == states[n][n] && states[n][j] == states[n][n])
                    reduced[i][j] += wrap(states[n][n]);

                if (states[i][n] == states[n][n] || states[n][j] == states[n][n])
                    reduced[i][j] += '+';
                else
                    reduced[i][j] += '*';
            }

            if (states[n][j] != states[n][n])
                reduced[i][j] += wrap(states[n][j]);

            reduced[i][j] = states[i][j] == 'phi' ? wrap(reduced[i][j]) : alternate(reduced[i][j], states[i][j]);
        }
    }
    return reduced;
}

function matching(x, open, close) {
    // Test if the parens are actually matching
    if ('(['.indexOf(x.charAt(open)) != -1 && ')]'.indexOf(x.charAt(close)) != -1) {
        var count = 0;
        for (var i = open; i <= close; i++) {
            if ('(['.indexOf(x.charAt(i)) != -1)
                count++;
            else if (')]'.indexOf(x.charAt(i)) != -1) {
                count--;

                if (count == 0)
                    return i == close;
            }
        }
    }

    return false;
}

function wrap(x) {
    if (x.length < 2 || matching(x, 0, x.length - 1))
        return x;
    return '(' + x + ')';
}

function optional(cond) {
    if (matching(cond, 0, cond.length - 2)) {
        var op = cond.charAt(cond.length - 1);
        if (op == '+')
            return cond.slice(0, -1) + '*';
        else if (op == '*' || op == '?')
            return cond;
    } else if (matching(cond, 0, cond.length - 1))
        return optional(cond.slice(1, -1));

    return wrap(cond) + '?';
}

function alternate(cond1, cond2) {
    cond2 = wrap(cond2);
    var index = cond1.indexOf(cond2);
    var len = cond2.length;
    var cond = '';

    if (index == 0) {
        var op = cond1.charAt(len);
        if (op == '*')
            cond = cond2 + '+' + optional(cond1.slice(len));
        else if (op == '+')
            cond = cond1;
        else 
            cond = cond2 + optional(cond1.slice(len));
    } else if (index == cond1.length - len)
        cond = optional(cond1.slice(0, index)) + cond2;
    else if (cond1.length == 1 && cond2.length == 1)
        cond = '[' + cond1 + cond2 + ']';
    else
        cond = cond1 + '|' + cond2;

    return wrap(cond);
}