As mentioned initially, I have only worked with positive integers from 1 to 360. This set yields a subset of 109 positive integers that do not produce a prime from the sum (

*P + i)* for any of the first 500,000 primes. The remaining 251 will produce a prime from this sum

*at least once.*
Creating a grid with 1...360 horizontally and the primes vertically, showing a black dot where the sum produces another prime, results in an intriguing pattern, indicating that some numbers

*i* yield a prime from the sum

*(P + i)* at a much higher percentage than others.

Of course, I would never rule out that I have made a mistake somewhere.

I'd love to dig deeper to see if there is a reoccurring pattern, but checking 1...360 for the first 100,000 primes already takes 7 minutes. If anyone has a very fast primality checker written in PHP that they'd like to share...

The subset of 109 positive integers:

7 13 19 23 25 31 33 37 43 47 49 53 55 61 63 67 73 75 79 83 85 89 91 93 97 103 109 113 115 117 119 121 123 127 131 133 139 141 143 145 151 153 157 159 163 167 169 173 175 181 183 185 187 193 199 201 203 205 207 211 213 215 217 219 223 229 233 235 241 243 245 247 251 253 257 259 263 265 271 273 277 283 285 287 289 293 295 297 299 301 303 307 313 317 319 321 323 325 327 331 333 337 339 341 343 349 353 355 359