In 1833, Heinrich Scherk conjectured that every prime of odd rank (accepting 1 as prime) can be composed by adding and subtracting all the smaller primes, each taken once. For instance, 13 is the 7th prime and 13 = 1 + 2 - 3 - 5 + 7 + 11.
Scherk's conjecture was proved to be true by J. L. Brown, Jr. in 1967. Information from FutilityCloset.
This calculator will produce the formula of smaller primes that produces any prime number input. It will also tell you if the 'prime' entered is not a prime. The first 100 primes are listed as examples to try with the calculator.
The first 50 prime numbers of odd rank: 3, 7, 13, 19, 29, 37, 43, 53, 61, 71, 79, 89, 101, 107, 113, 131, 139, 151, 163, 173, 181, 193, 199, 223, 229, 239, 251, 263, 271, 281, 293, 311, 317, 337, 349, 359, 373, 383, 397, 409, 421, 433, 443, 457, 463, 479, 491, 503, 521, 541
The algorithm for finding the formula will proceed as follows. All of the smaller primes will be summed. Assuming this sum is larger than the prime, half of the difference needs to be subtracted. The largest prime smaller than half of the difference (or equal) will be subtracted. This process is repeated until the difference is matched. If not matched the process will go back a step and try a different prime to subtract.
The limit is based solely on your patience. A limit of 100,000,000 takes about 20 seconds to calculate. Of course, the largest possible factorion is less than 99,999,999 as 8 X 9! = 2,903,040. Further digits do not grow fast enough as 9! < 1,000,000.
Reload the browser window/tab between calculations if you don't want successive calculations appended.