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The density of primes end with 1 is relatively low (< 1/4), but the density of primes end with 5, 7 and E are nearly equal (all are a little more than 1/4). (i.e. for a given natural number ''N'', the number of primes end with 1 less than ''N'' is usually smaller than the number of primes end with 5 (or 7, or E) less than ''N'') e.g. For all 1426 primes < 10000, there are 3E8 primes (2E.3%) end with 1, 410 primes (30.3%) end with 5, 412 primes (30.5%) end with 7, 406 primes (2E.E%) end with E. It is conjectured that for every natural number ''N'' β₯ 10, the number of primes end with 1 less than ''N'' is smaller than the number of primes end with 5 (or 7, or E) less than ''N''. (Note: the [[percentage]] in this section are also in dozenal, i.e. 20% means 0.2 or 20/100 = 1/6, 36% means 0.36 or 36/100 = 7/20, 58.7% means 0.587 or 587/1000) |
The density of primes end with 1 is relatively low (< 1/4), but the density of primes end with 5, 7 and E are nearly equal (all are a little more than 1/4). (i.e. for a given natural number ''N'', the number of primes end with 1 less than ''N'' is usually smaller than the number of primes end with 5 (or 7, or E) less than ''N'') e.g. For all 1426 primes < 10000, there are 3E8 primes (2E.3%) end with 1, 410 primes (30.3%) end with 5, 412 primes (30.5%) end with 7, 406 primes (2E.E%) end with E. It is conjectured that for every natural number ''N'' β₯ 10, the number of primes end with 1 less than ''N'' is smaller than the number of primes end with 5 (or 7, or E) less than ''N''. (Note: the [[percentage]] in this section are also in dozenal, i.e. 20% means 0.2 or 20/100 = 1/6, 36% means 0.36 or 36/100 = 7/20, 58.7% means 0.587 or 587/1000) |
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β | 13665 is the smallest prime ''p'' such that the number of primes end with 1 or 5 β€ ''p'' is more than the number of primes end with 3, 7 or E β€ ''p'' ( |
+ | 13665 is the smallest prime ''p'' such that the number of primes end with 1 or 5 β€ ''p'' is more than the number of primes end with 3, 7 or E β€ ''p'' (see <ref>[https://oeis.org/A007350]</ref>, of course, 3 is the only prime ends with 3). Besides, 9X03693X831 is the smallest prime ''p'' such that the number of primes end with 1 or 7 β€ ''p'' is more than the number of primes end with 2, 5 or E β€ ''p'' (see <ref>[https://oeis.org/A007352]</ref>, of course, 2 is the only prime ends with 2). Besides, 2E69E is the smallest prime ''p'' such that the number of primes end with 1 or E β€ ''p'' is more than the number of primes end with 5 or 7 β€ ''p'' (of course, 2 is the only prime ends with 2, and 3 is the only prime ends with 3). '''Question: What is the smallest prime ''p'' such that the number of primes end with 1 β€ ''p'' is more than the number of primes end with ''d'' β€ ''p'' for at least one of ''d'' = 5, 7 or E?''' |
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