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| Figure 1: permalink |
As I keep my tally of the days that have passed since my birth, I've noticed that lately the primes have become a little thin. To be specific, on August 26th 2015 I celebrated prime day 24251. It was to be prime day 24281 (yesterday) before the drought of primes would be broken, a gap of 30 days. The next prime day is now 24317, a gap of 36.
The bottom half of Figure 1 shows where the maximum gaps occur for the smallest prime numbers. For example, after prime 9551 there is a gap of 36 (so the next prime is 9587) and that was the first time a gap of that magnitude occurred.
Of course the average gap to the next prime for any prime p is given approximately by ln(p) and ln(24281)= 10.097 or just 10 for short. The gap increases slowly as the numbers become larger but even at 242810 for example, the log returns 12.400. Well, the prime days will flow again but now the next one is five weeks away. So it goes.
The top half of Figure 1 is the SageMath code used to make a dictionary that records the record gaps up to one hundred million (100,000,000) mark and output the result in tabular form.
If we want to search for occurrences of specific gaps, Figure 2 shows the primes (up to 30,000) that are followed by gaps of 36. The first occurrence is 9551 and a later one is 24281 as mentioned earlier.
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| Figure 2: permalink |
on December 18th 2020
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