# Is $48^p-47^p$ ever prime?

I was examining primes q that are $q=(n+1)^p-n^p$ with p also prime. There seems to be more than one solution for most n. In the table below all the first solutions of p up to 2000 and for n up to 200: \begin{array}{||r|r|||r|r|||r|r|||r|r||} \hline \hline n & p & n & p & n & p & n & p \\ \hline \hline 1 & 3 &51 & 29 &101 & 5 &151 & 59\\ \hline 2 & 3 &52 & 3 &102 & 13 &152 & 1583\\ \hline 3 & 3 &53 & 479 &103 & 7 &153 & 3\\ \hline 4 & 3 &54 & 5 &104 & 5 &154 & 43\\ \hline 5 & 5 &55 & 3 &105 & 3 &155 & 7\\ \hline 6 & 3 &56 & 19 &106 & - &156 & 3\\ \hline 7 & 7 &57 & 5 &107 & 59 &157 & 3\\ \hline 8 & 7 &58 & 3 &108 & 3 &158 & 3\\ \hline 9 & 3 &59 & - &109 & 5 &159 & 7\\ \hline 10 & 3 &60 & - &110 & 5 &160 & 13\\ \hline 11 & 3 &61 & 17 &111 & 113 &161 & 7\\ \hline 12 & 17 &62 & 3 &112 & 5 &162 & 389\\ \hline 13 & 3 &63 & 3 &113 & - &163 & 47\\ \hline 14 & 3 &64 & 5 &114 & 139 &164 & 3\\ \hline 15 & 43 &65 & 7 &115 & 269 &165 & 3\\ \hline 16 & 5 &66 & 3 &116 & - &166 & 13\\ \hline 17 & 3 &67 & 3 &117 & 7 &167 & 167\\ \hline 18 & 1607 &68 & 17 &118 & 13 &168 & 11\\ \hline 19 & 5 &69 & 11 &119 & 3 &169 & 17\\ \hline 20 & 19 &70 & 47 &120 & 5 &170 & 3\\ \hline 21 & 127 &71 & 61 &121 & 5 &171 & 3\\ \hline 22 & 229 &72 & 19 &122 & 7 &172 & 3\\ \hline 23 & 3 &73 & 23 &123 & 3 &173 & 19\\ \hline 24 & 3 &74 & 3 &124 & - &174 & 5\\ \hline 25 & 3 &75 & 5 &125 & 3 &175 & 3\\ \hline 26 & 13 &76 & 19 &126 & 41 &176 & 37\\ \hline 27 & 3 &77 & 7 &127 & 59 &177 & 269\\ \hline 28 & 3 &78 & 5 &128 & 3 &178 & 5\\ \hline 29 & 149 &79 & 7 &129 & 3 &179 & 3\\ \hline 30 & 3 &80 & 3 &130 & 1499 &180 & 19\\ \hline 31 & 5 &81 & 3 &131 & 101 &181 & 43\\ \hline 32 & 3 &82 & 331 &132 & 167 &182 & 43\\ \hline 33 & 23 &83 & 41 &133 & - &183 & 11\\ \hline 34 & 3 &84 & 179 &134 & 7 &184 & 3\\ \hline 35 & 5 &85 & 5 &135 & 7 &185 & 3\\ \hline 36 & 83 &86 & 3 &136 & 3 &186 & 3\\ \hline 37 & 3 &87 & 5 &137 & - &187 & -\\ \hline 38 & 3 &88 & 3 &138 & 113 &188 & 13\\ \hline 39 & 37 &89 & 109 &139 & - &189 & 41\\ \hline 40 & 7 &90 & 3 &140 & 3 &190 & 53\\ \hline 41 & 3 &91 & 3 &141 & 7 &191 & 3\\ \hline 42 & 3 &92 & 17 &142 & 3 &192 & 5\\ \hline 43 & 37 &93 & 3 &143 & - &193 & 3\\ \hline 44 & 5 &94 & 61 &144 & 5 &194 & -\\ \hline 45 & 3 &95 & 3 &145 & 7 &195 & 3\\ \hline 46 & 5 &96 & 1307 &146 & 13 &196 & 3\\ \hline 47 & - &97 & 7 &147 & 3 &197 & 5\\ \hline 48 & 3 &98 & 709 &148 & 5 &198 & -\\ \hline 49 & 3 &99 & 5 &149 & 271 &199 & 31\\ \hline 50 & 7 &100 & 43 &150 & 13 &200 & 59\\ \hline \hline \end{array} 47 is the first n that has no solution for $p<2000$. I tried n=47 for higher p but still no solution for $p<10000$

Question: Is there a way to prove that $q=48^p-47^p$ is never prime? After factoring $48^p-47^p$ for several p, I found the smallest prime factor is always $\equiv 1 \pmod {p}$ but I don't see how this helps.

• Sorry but where are the mathematics in "Q.: Is there some integer p such that property P holds?" "A.: Yes, because the OEIS says so"? – Did Apr 12 '17 at 13:58
• @Did OEIS says so because people have calculated to that number and found that for p=58543, the property P holds. – DHMO Apr 12 '17 at 14:09
• @DHMO No kidding? Of course they have. And where should I feel enlightened (mathematically speaking) by this question and by your (technically accurate) answer? – Did Apr 12 '17 at 14:18
• @Did Nowhere at all. – DHMO Apr 12 '17 at 14:20
• Still formulated otherwise: the question is tagged (number-theory), I see the numbers but not the theory. – Did Apr 12 '17 at 14:23

According to A125713, $n=58543$ corresponds to a prime.
• @Shaun The question is whether $48^p-47^p$ can be prime. I gave an answer: "yes, when $p=58543$". – DHMO Apr 12 '17 at 14:05