\( \textbf{primes with the sums of all pairs of successive digits square} \)
13, 31, 79, 97, 101, 109, 131, 181, 227, 313, 401, 409, 631, 727, 797, 881, 1009, 1013, 1097, 2797, 3109, 3181, 3631, 4001, 4013, 7901, 8101, 9001, 9013, 10009, 10181, 10909, 10979, 13109, 18131, 18181, 22279, 22727, 27901, 31013, 36313
If a number is prime and the sums of all pairs of successive digits are \( \textbf{prime} \) as well then we find that there are 160 numbers that qualify in the range up 40000 (permalink):
\( \textbf{primes with the sums of all pairs of successive digits prime} \)
11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 149, 167, 211, 307, 347, 349, 383, 389, 503, 521, 523, 743, 761, 929, 941, 947, 983, 1123, 1129, 2029, 2111, 2129, 2141, 2143, 2161, 2341, 2347, 2383, 2389, 2503, 2521, 3023, 3203, 4111, 4129, 4349, 4703, 4943, 5021, 5023, 6121, 6143, 6521, 6529, 6703, 6761, 7411, 8329, 8389, 8521, 8923, 8929, 8941, 9203, 11149, 11161, 11411, 12143, 12149, 12161, 12323, 12329, 12343, 12347, 12503, 12583, 12589, 12923, 12941, 12983, 14143, 14149, 14303, 14321, 14323, 14341, 14347, 14389, 14741, 14747, 14767, 14923, 14929, 14947, 14983, 16111, 16141, 16529, 16561, 16567, 16703, 16741, 16747, 20323, 20341, 20347, 20389, 20507, 20521, 20707, 20743, 20747, 20749, 21121, 21143, 21149, 21211, 21611, 23021, 23029, 23203, 29207, 29411, 30203, 30211, 30307, 30323, 30341, 30347, 30389, 30529, 30703, 30707, 32029, 32141, 32143, 32303, 32321, 32323, 32341, 32503, 32507, 32561, 32941, 32983, 34123, 34129, 34141, 34147, 34303, 34703, 34747, 34949, 38303, 38321, 38329, 38561, 38567, 38921, 38923
Let's take the final number, \( \textbf{38923} \), above as an example. We have:$$ \begin{align} 3 + 8 &=11\\8 + 9 &= 17 \\ 9+2 &=11\\2+3 &=5 \end{align}$$We can add an additional constraint here and that is that the \( \textbf{first and last digits} \) be considered adjacent and prime as well. In this case, the suitable numbers in the range up to 40000 shrink to 60. These numbers belong to OEIS A086244 (permalink):$$ \begin{align} &\textbf{primes with the sums of all pairs of successive } \\ &\textbf{digits prime as well as sums of first and last digits} \end{align}$$11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 211, 2029, 2111, 2129, 2141, 2143, 2161, 2341, 2383, 2389, 2503, 2521, 4111, 4129, 4349, 4703, 4943, 6121, 6521, 6761, 8329, 8389, 8923, 8929, 11161, 11411, 12161, 12941, 14321, 14341, 14741, 16111, 16141, 16561, 16741, 20323, 20341, 20389, 20521, 20743, 20749, 21121, 21143, 21149, 21211, 21611, 23021, 23029, 23203, 29411
Let's take the final number in the previous list, \( \textbf{29411} \), and show that it satisfies the criteria:$$ \begin{align} 2 + 9 &= 11 \\ 9 + 4 &= 13 \\ 4 + 1 &= 5 \\ 1 + 1 &= 2 \\ 2 + 1 &= 3 \end{align}$$We can also consider primes where the absolute values of \( \textbf{differences} \) between successive pairs of digits are prime. There are 272 of these in the range up to 40000. They constitute OEIS A087593 (permalink):$$ \begin{align} &\textbf{primes with the absolute differences} \\ &\textbf{of all pairs of successive digits prime} \end{align} $$13, 29, 31, 41, 47, 53, 61, 79, 83, 97, 131, 149, 163, 181, 241, 257, 307, 313, 353, 383, 461, 463, 479, 503, 613, 631, 641, 647, 683, 727, 757, 797, 853, 857, 863, 929, 941, 947, 1303, 1307, 1361, 1381, 1427, 1429, 1613, 1697, 1831, 1861, 2027, 2029, 2053, 2503, 2531, 2579, 2707, 2729, 2741, 2749, 2753, 2797, 2927, 2963, 2969, 3079, 3163, 3169, 3181, 3527, 3529, 3581, 3583, 3613, 3631, 3697, 3853, 3863, 4241, 4253, 4297, 4649, 4703, 4729, 4969, 5279, 5297, 5303, 5381, 5741, 5749, 5813, 5857, 5861, 5869, 6131, 6163, 6353, 6361, 6427, 6469, 6857, 6863, 6869, 6947, 6949, 6961, 7027, 7057, 7079, 7207, 7247, 7253, 7297, 7507, 7529, 7583, 7927, 7949, 7963, 8147, 8161, 8353, 8363, 8369, 8527, 8581, 8641, 8647, 8681, 9203, 9241, 9257, 9413, 9461, 9463, 9479, 9497, 9613, 9631, 9649, 9697, 9749, 13147, 13163, 13183, 13613, 13649, 13681, 13697, 13831, 14149, 14207, 14249, 14683, 14741, 14747, 14753, 14797, 14929, 14947, 14969, 16141, 16183, 16361, 16363, 16369, 16381, 16427, 16831, 16927, 16963, 16979, 18131, 18149, 18169, 18181, 18307, 18313, 18353, 18503, 18583, 20249, 20297, 20353, 20357, 20369, 20507, 20707, 20747, 20749, 20753, 24169, 24181, 24203, 24247, 24631, 24683, 24697, 24749, 24979, 25031, 25057, 25247, 25253, 25303, 25307, 25357, 25703, 25741, 25747, 27031, 27241, 27253, 27427, 27479, 27527, 27529, 27581, 27583, 27941, 27947, 27961, 29207, 29297, 29429, 29641, 29683, 29741, 29753, 30203, 30241, 30253, 30307, 30313, 30529, 30703, 30707, 30727, 30757, 31307, 31357, 31469, 31649, 35027, 35053, 35257, 35279, 35353, 35363, 35381, 35729, 35747, 35753, 35797, 35831, 35863, 35869, 36131, 36161, 36307, 36313, 36353, 36383, 36469, 36479, 36497, 36857, 36929, 36947, 36979, 38149, 38183, 38303
Let's take the last number, \( \textbf{38303}\), in the list above. We have:$$ \begin{align} |3-8|=5 \\ |8 - 3|=5 \\ |3-0|=3 \\ |0-3|=3 \end{align}$$There are all sorts of variations on this theme (the properties of pairs of adjacent digits) and so another approach is to consider the squares of the digits. Let's require that the sums of squares of adjacent digits be prime. We find that there are 71 numbers that qualify in the range up to 40000. These are (permalink):$$ \begin{align} \textbf{primes with the sums of all pairs}\\ \textbf{of successive digits squared prime} \end{align} $$11, 23, 41, 61, 83, 127, 149, 211, 383, 521, 523, 541, 587, 727, 787, 941, 1123, 2111, 2141, 2161, 2383, 2521, 2549, 4111, 4127, 4523, 4549, 4561, 4583, 6121, 6521, 7211, 8387, 8521, 8527, 8783, 11149, 11161, 11411, 12149, 12161, 12323, 12527, 12541, 12583, 12721, 14149, 14549, 14561, 16111, 16127, 16141, 16561, 21121, 21149, 21211, 21611, 25411, 27211, 32141, 32321, 32323, 32327, 32561, 32587, 32783, 38321, 38327, 38561, 38723, 38783
Let's take the last number, \( \textbf{38783} \), as an example:$$ \begin{align} 3^2+8^2 &= 9 + 64 =73 \\ 8^2+7^2 &= 64 +49 = 113 \\ 7^2+8^2 &= 49+64 = 113 \\ 8^2+3^2 &= 64 + 9 = 73 \end{align}$$
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