Primes from Primes

I've begun reading "The Man Who Loved Only Numbers" by Paul Hoffman, a biography of Paul Erdös. Figure 1 shows the front cover of the book. It motivated me to be a little more energetic in my daily number analysis at least for today because today was a prime day.

Figure 1

THE STORY OF PAUL ERDÖS AND THE SEARCH FOR MATHEMATICAL TRUTH

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By that I mean I turned a prime number of days old, specifically 26371. Initially, I'd found that this number was a member of OEIS A255543:

A255543

Unlucky array: Row \(n\) consists of unlucky numbers removed at the stage \(n\) of Lucky sieve.

Figure 2, taken from the OEIS entry comments, shows what is meant by this:

Figure 2

Looking at the first row, it can seen that 2 and all multiples of 2 are removed. In the second row, every third remaining number is removed and so on for successive rows. 26371 lies in the 29th row that lists all the numbers removed when every 29th number is struck off. This was interesting but didn't relate to any specific properties of 26371 as a prime number. A little more research, motivated by Erdös's indefatigable research, led me to OEIS A249350:

A249350

Prime numbers Q such that the concatenation Q, 6, Q is prime.

As a member of this sequence, 26371 has the property that 26371626371 is a prime number. Up to 26371, the list of such primes is:

[13, 23, 29, 41, 53, 59, 71, 73, 89, 107, 149, 167, 173, 197, 239, 241, 257, 293, 349, 379, 383, 397, 439, 457, 461, 479, 503, 521, 547, 569, 607, 617, 631, 643, 677, 691, 727, 733, 757, 821, 887, 919, 941, 947, 953, 967, 1051, 1061, 1069, 1097, 1103, 1187, 1213, 1217, 1237, 1279, 1297, 1373, 1399, 1409, 1423, 1433, 1451, 1453, 1471, 1483, 1499, 1567, 1609, 1619, 1621, 1667, 1709, 1721, 1723, 1783, 1787, 1789, 1861, 1867, 1889, 1913, 1993, 1997, 2011, 2017, 2029, 2063, 2099, 2113, 2251, 2269, 2273, 2357, 2393, 2441, 2473, 2503, 2557, 2609, 2647, 2657, 2659, 2687, 2699, 2711, 2713, 2777, 2843, 2897, 2927, 2953, 3037, 3061, 3079, 3137, 3217, 3271, 3323, 3343, 3499, 3511, 3527, 3547, 3557, 3593, 3631, 3659, 3673, 3733, 3779, 3851, 3911, 4051, 4093, 4129, 4241, 4243, 4253, 4327, 4339, 4373, 4391, 4457, 4493, 4519, 4561, 4583, 4597, 4603, 4639, 4643, 4663, 4723, 4787, 4789, 4801, 4813, 4877, 4933, 4951, 4967, 5011, 5023, 5051, 5179, 5209, 5333, 5413, 5527, 5557, 5647, 5807, 5851, 5857, 5867, 5903, 6067, 6113, 6173, 6199, 6311, 6353, 6379, 6553, 6571, 6659, 6781, 6827, 6841, 6871, 6949, 6997, 7013, 7079, 7151, 7177, 7193, 7237, 7349, 7393, 7459, 7481, 7523, 7529, 7541, 7559, 7573, 7589, 7607, 7621, 7673, 7687, 7793, 7817, 7823, 7841, 7867, 7873, 7907, 8087, 8093, 8101, 8209, 8317, 8369, 8387, 8419, 8429, 8447, 8461, 8467, 8573, 8623, 8647, 8677, 8681, 8699, 8741, 8779, 8803, 8821, 8861, 8971, 8999, 9013, 9059, 9133, 9137, 9181, 9199, 9239, 9283, 9337, 9343, 9419, 9431, 9461, 9473, 9511, 9533, 9539, 9629, 9767, 9883, 10103, 10133, 10223, 10357, 10487, 10559, 10691, 10729, 10847, 10853, 10909, 10957, 10979, 11083, 11093, 11117, 11159, 11177, 11243, 11273, 11321, 11329, 11369, 11393, 11471, 11483, 11489, 11491, 11813, 11887, 12007, 12049, 12119, 12211, 12239, 12253, 12281, 12289, 12379, 12413, 12479, 12517, 12527, 12553, 12647, 12703, 12721, 12889, 12919, 13003, 13037, 13043, 13147, 13163, 13171, 13381, 13499, 13679, 13757, 13877, 14009, 14051, 14057, 14071, 14081, 14207, 14423, 14449, 14627, 14723, 14767, 14813, 14869, 14879, 14939, 15031, 15061, 15101, 15131, 15173, 15193, 15299, 15373, 15377, 15383, 15541, 15559, 15629, 15643, 15649, 15787, 15877, 15919, 15923, 16189, 16333, 16339, 16361, 16427, 16487, 16529, 16607, 16649, 16763, 16871, 16903, 16931, 17011, 17021, 17029, 17033, 17077, 17137, 17419, 17483, 17729, 17747, 17749, 17851, 17903, 17921, 17957, 17981, 18041, 18049, 18169, 18257, 18397, 18413, 18517, 18541, 18583, 18671, 18691, 18701, 18719, 18749, 18757, 18803, 18973, 19069, 19211, 19213, 19289, 19379, 19463, 19471, 19489, 19603, 19819, 19843, 19861, 19919, 20071, 20101, 20147, 20261, 20297, 20399, 20443, 20681, 20707, 20731, 20849, 20897, 20921, 20939, 21001, 21011, 21059, 21089, 21121, 21163, 21169, 21221, 21227, 21313, 21341, 21401, 21407, 21467, 21523, 21569, 22109, 22129, 22171, 22247, 22349, 22639, 22643, 22741, 22769, 22787, 22811, 22961, 23027, 23041, 23143, 23201, 23203, 23339, 23357, 23369, 23459, 23537, 23627, 23629, 23747, 23767, 23819, 23857, 23879, 23887, 24007, 24019, 24029, 24061, 24097, 24151, 24391, 24407, 24421, 24683, 24767, 24851, 24953, 25033, 25147, 25253, 25321, 25439, 25643, 26119, 26189, 26237, 26357, 26371]

26371 is the 2897th prime and the primes listed above total 502. This means that of all the primes up 26371, 502 or about 16.8% generate a new prime according the Q + 6 + Q concatenation. I wondered what numbers arise when the digits 1, 2, 3, 4, 5, 7, 8 and 9 are used instead. Inserting 0 between the two primes cannot produce a prime because the resulting concatenated number is always divisible by Q. Here are the figures for the digits from 1 to 9:

1     278

2     238

3     528

4     242

5     258

6     502

7     296

8     247

9     512

total is 3101

It can be seen that the record is held by the digit 3, although 6 and 9 are close behind. Well back however, are the digits 1, 2, 4, 5, 7 and 8. I thought I'd extend this to the first one million primes and Figure 3 shows the results obtained:

Figure 3

The proportions remain about the same with the exception of the digit 7. Figure 4 shows a table summarising the results:

Figure 4

Why do the digits 3, 6 and 9 produce about twice as many primes as the digits 1, 2, 4, 5 and 8? Why does the digit 7 produce significantly fewer primes that 1, 2, 4, 5 and 8? These are questions that I don't know the answer to but I'm keen to investigate.

One doesn't have to stop at the digit 9. What happens for the digits 10 to 19? Figure 5 tells the tale.

Figure 5

Figure 6 shows the same results in tabular form. It's clear that the multiples of 3 (12, 15 and 18) always win the day and with consistent frequency. The digits 10, 16 and 17 produce about half as many primes as their multiple of 3 counterparts, while 11, 13, 14 and 19 produce less than a third of even this number.

Figure 6

One might surmise that the frequency for multiples of 3 remains relatively constant as we investigate higher digits. After all, the numbers for 3, 6, 9, 12, 15 and 18 have been quite consistent. However, 21 = 3 x 7 breaks the pattern. See Figure 7.

Figure 7

The figure for 21 is not as low as for 22, 26 and 28 but it significantly lower than even the figures for 20, 23, 25 and 29. Figure 8 presents the results in tabular form.

Figure 8

Multiples of 7, 11 and 13 seem to produce far fewer primes when concatenated using Q + digit + Q. Figure 9 provides an overview of the digits from 1 to 99:

Figure 9

Clearly, there is more to be discovered here but I'll finish up at this point. What this post teaches us more than anything else is to not let a good prime go to waste and to be a little more energetic in my investigations.