Regexes where it is not known whether they match anything

goldbachregex.js

// I claim that the following regular expression should match a string if and only if
// it is composed of only the character "x", and
// its length is an even number greater than two which cannot be expressed as the sum of two prime numbers
// that is, if the length of the string is a counterexample to Goldbach's conjecture.
const r = /^(?!^(x*(?<!^(?:x?|\2+(xx+))))((?!(?:x?|(xx+?)\4+)$)x*)$)xx(xx)+$/;

console.log("start");
for (let i = 0; i < 1000; i++) {    
    const s = "x".repeat(i);
        if (r.test(s)) console.log(i)
}
console.log("done");

// well, that proves it doesn't match any lengths less than 1000.  
// Of course any counterexample to Goldbach's conjecture would be much larger than the maximum possible string 
// length representable in JavaScript, so I guess we can be sure that r matches no possible inputs

// ---------------------

// In case you doubt the above claim, this related regular expression *matches* all even numbers that can be represented as the sum of
// two primes, and we can use the capture groups to show the primes
const goldbachsRegex = /^(?=^(x*(?<!^(?:x?|\2+(xx+))))((?!(?:x?|(xx+?)\4+)$)x*)$)xx(xx)+$/

const results: string[] = [];

for (let i = 0; i < 500; i++) {
  const s = "x".repeat(i).match(goldbachsRegex);
  if (s) results.push(`${s[0].length}=${s[1].length}+${s[3].length}`);
}

console.log(results.join("; "));

/* 
4=2+2; 6=3+3; 8=5+3; 10=7+3; 12=7+5; 14=11+3; 16=13+3; 18=13+5; 20=17+3; 22=19+3; 24=19+5; 
 26=23+3; 28=23+5; 30=23+7; 32=29+3; 34=31+3; 36=31+5; 38=31+7; 40=37+3; 42=37+5; 44=41+3;
  46=43+3; 48=43+5; 50=47+3; 52=47+5; 54=47+7; 56=53+3; 58=53+5; 60=53+7; 62=59+3; 64=61+3; 
  66=61+5; 68=61+7; 70=67+3; 72=67+5; 74=71+3; 76=73+3; 78=73+5; 80=73+7; 82=79+3; 84=79+5; 
  86=83+3; 88=83+5; 90=83+7; 92=89+3; 94=89+5; 96=89+7; 98=79+19; 100=97+3; 102=97+5; 
  104=101+3; 106=103+3; 108=103+5; 110=107+3; 112=109+3; 114=109+5; 116=113+3; 118=113+5; 
  120=113+7; 122=109+13; 124=113+11; 126=113+13; 128=109+19; 130=127+3; 132=127+5; 134=131+3; 
  136=131+5; 138=131+7; 140=137+3; 142=139+3; 144=139+5; 146=139+7; 148=137+11; 150=139+11; 
  152=149+3; 154=151+3; 156=151+5; 158=151+7; 160=157+3; 162=157+5; 164=157+7; 166=163+3; 
  168=163+5; 170=167+3; 172=167+5; 174=167+7; 176=173+3; 178=173+5; 180=173+7; 182=179+3; 
  184=181+3; 186=181+5; 188=181+7; 190=179+11; 192=181+11; 194=191+3; 196=193+3; 198=193+5; 
  200=197+3; 202=199+3; 204=199+5; 206=199+7; 208=197+11; 210=199+11; 212=199+13; 214=211+3; 
  216=211+5; 218=211+7; 220=197+23; 222=211+11; 224=211+13; 226=223+3; 228=223+5; 230=227+3; 
  232=229+3; 234=229+5; 236=233+3; 238=233+5; 240=233+7; 242=239+3; 244=241+3; 246=241+5; 
  248=241+7; 250=239+11; 252=241+11; 254=251+3; 256=251+5; 258=251+7; 260=257+3; 262=257+5; 
  264=257+7; 266=263+3; 268=263+5; 270=263+7; 272=269+3; 274=271+3; 276=271+5; 278=271+7; 
  280=277+3; 282=277+5; 284=281+3; 286=283+3; 288=283+5; 290=283+7; 292=281+11; 294=283+11; 
  296=293+3; 298=293+5; 300=293+7; 302=283+19; 304=293+11; 306=293+13; 308=277+31; 310=307+3; 
  312=307+5; 314=311+3; 316=313+3; 318=313+5; 320=317+3; 322=317+5; 324=317+7; 326=313+13; 
  328=317+11; 330=317+13; 332=313+19; 334=331+3; 336=331+5; 338=331+7; 340=337+3; 342=337+5; 
  344=337+7; 346=317+29; 348=337+11; 350=347+3; 352=349+3; 354=349+5; 356=353+3; 358=353+5; 
  360=353+7; 362=359+3; 364=359+5; 366=359+7; 368=349+19; 370=367+3; 372=367+5; 374=367+7; 
  376=373+3; 378=373+5; 380=373+7; 382=379+3; 384=379+5; 386=383+3; 388=383+5; 390=383+7; 
  392=389+3; 394=389+5; 396=389+7; 398=379+19; 400=397+3; 402=397+5; 404=401+3; 406=401+5; 
  408=401+7; 410=397+13; 412=409+3; 414=409+5; 416=409+7; 418=401+17; 420=409+11; 422=419+3; 
  424=421+3; 426=421+5; 428=421+7; 430=419+11; 432=421+11; 434=431+3; 436=433+3; 438=433+5; 
  440=433+7; 442=439+3; 444=439+5; 446=443+3; 448=443+5; 450=443+7; 452=449+3; 454=449+5; 
  456=449+7; 458=439+19; 460=457+3; 462=457+5; 464=461+3; 466=463+3; 468=463+5; 470=467+3; 
  472=467+5; 474=467+7; 476=463+13; 478=467+11; 480=467+13; 482=479+3; 484=479+5; 486=479+7;
  488=457+31; 490=487+3; 492=487+5; 494=491+3; 496=491+5; 498=491+7
  */