Luhn Primes

Today, the day before my 72nd birthday, I turned 26297 days old. Now 26297 is a prime number and in my search to find some interesting properties relating to this number, I stumble upon a book titled "Various Arithmetic Functions and their Applications" by Octavian Cira and Florentin Smarandache. What I found was that:$$26297^4+79262^4=39947578799194466417$$The rather large number on the right hand side of the above equation is also prime and this qualifies 26297 as a Luhn prime of the fourth order. However, I'm getting ahead of myself. Let's clarify what a Luhn prime is by quoting from the aforementioned book:

The number 229 is the smallest prime which summed with its inverse gives also a prime. Indeed, 1151 is a prime, and 1151 = 229 + 922. The first to note this special property of 229, on the website Prime Curios, was Norman Luhn (9 Feb. 1999), [Luhn, 2013, Caldwell and Honacher Jr., 2014].

These Luhn primes of the first rank, although simply called Luhn primes, constitute OEIS A061783:

A061783

Luhn primes: primes \(p\) such that \(p + (p \text{ reversed }) \) is also a prime.

 The sequence runs:

229, 239, 241, 257, 269, 271, 277, 281, 439, 443, 463, 467, 479, 499, 613, 641, 653, 661, 673, 677, 683, 691, 811, 823, 839, 863, 881, 20011, 20029, 20047, 20051, 20101, 20161, 20201, 20249, 20269, 20347, 20389, 20399, 20441, 20477, 20479, 20507, ...

The authors of the book ask the question as to whether there are Luhn primes of second rank? Their answer is yes, indeed. 23 is a Luhn prime number of second rank because 1553 is a prime and we have \(1553 = 23^2 + 32^2\). The sequence of such numbers forms OEIS A304390:

A304390

Prime numbers \(p\) such that \(p^2\) + \( (p \text{ reversed })^2 \) is also prime.

This sequence runs:

23, 41, 227, 233, 283, 401, 409, 419, 421, 461, 491, 499, 823, 827, 857, 877, 2003, 2083, 2267, 2437, 2557, 2593, 2617, 2633, 2677, 2857, 2887, 2957, 4001, 4021, 4051, 4079, 4129, 4211, 4231, 4391, 4409, 4451, 4481, 4519, 4591, 4621, 4639, 4651, 4871, 6091, 6301, 6329, 6379, 6521, 6529, 6551, ...

Interestingly there doesn't seem to be any Luhn primes of the third rank. However, as we've seen Luhn primes of the fourth order are possible and my diurnal age, 26297, is an example. These particular primes are not listed in the OEIS but, copying from the book, here is a list up to 26297:

23, 43, 47, 211, 233, 239, 263, 419, 431, 487, 491, 601, 683, 821, 857, 2039, 2063, 2089, 2113, 2143, 2203, 2243, 2351, 2357, 2377, 2417, 2539, 2617, 2689, 2699, 2707, 2749, 2819, 2861, 2917, 2963, 4051, 4057, 4127, 4129, 4409, 4441, 4481, 4603, 4679,  4733, 4751, 4951, 4969, 4973, 6053, 6257, 6269, 6271, 6301, 6311, 6353, 6449, 6547, 6551, 6673, 6679, 6691, 6803, 6869, 6871, 6947, 6967, 8081, 8123, 8297, 8429, 8461, 8521, 8543, 8627, 8731, 8741, 8747, 8849, 8923, 8951, 8969, 20129, 20149, 20177, 20183, 20903, 20921, 21017, 21613, 21661, 21727, 22073, 22133, 22171, 22817, 22853, 22877, 23531, 23767, 23827, 24251, 24421, 24481, 25307, 25321, 25343, 26171, 26267, 26297, ...

Thus they have the property that:

OEIS candidate?

Prime numbers \(p\) such that \(p^4\) + \( (p \text{ reversed })^4 \) is also prime.

The authors make the comment that:

Up to \(3 \times 10^4\), the numbers: 23, 233, 419, 491, 857, 2617, 4051, 4129, 4409, 4481, 6301, 6551, 6871, 8543, 21727, 21803, 21937, 22133, 23227, 23327, 24527, 28297, 29063 are Luhn prime numbers of 2nd and 4th rank. The question of whether there are Luhn primes of rank higher than 4 is posed as a question in the book but is not answered.

The book will continue to prove useful in the future I'm sure. Here is some information about the book and its contents:

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes, squares, cubes, factorials, almost primes, mobile periodicals, functions, tables, prime square factorial bases, generalised factorials, generalised palindromes and so on, have been extracted from the Archives of American Mathematics (University of Texas at Austin) and Arizona State University (Tempe): "The Florentin Smarandache papers" special collections, and Arhivele Statului (Filiala Vâlcea & Filiala Dolj, Romania).

This book was born from the collaboration of the two authors, which started in 2013. The first common work was the volume "Solving Diophantine Equations", published in 2014. The contribution of the authors can be summarised as follows: Florentin Smarandache came with his extraordinary ability to propose new areas of study in number theory, and Octavian Cira - with his algorithmic thinking and knowledge of Mathcad.