Friday, October 5

What else?

Last month's seven consecutive primes summing to a repunit prime has made it to a PrimePuzzles website, where my "what else?" is transmuted into Q1. I'll note here my own negative results.

The seven-consecutive-primes is the only solution that I found for R19 (the repunit consisting of 19 ones) up to 10000 consecutive primes. I found no solution for R23 up to 10000 consecutive primes. I found no solution for R317 up to 1000 consecutive primes. I found no solution for R1031 up to 500 consecutive primes. Finally, I can show that R49081 is not the sum of three consecutive primes:

Let k equal (R49081 minus 1) divided by 3. This integer is not prime. The largest prime less than k is ((10^49081-1)/9-1)/3-71387 and the smallest prime greater than k is ((10^49081-1)/9-1)/3+84453. These two primes are therefore consecutive and both are required to be part of the three consecutive primes that might sum to R49081. It's an easy matter now to determine that the third number needs to be ((10^49081-1)/9-1)/3-13065 which (alas) is between our two consecutive primes and is therefore composite.

By the way, if you think that the 155839 composites between the two (above) consecutive primes constitutes a big prime gap, you'd be wrong. The gap has a "merit" of only 1.379. A website that charts large prime gaps won't even consider gaps with a merit less than 10.

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