Is there a prime of the form $a^b+b^c+c^d$ with consecutive primes $a,b,c,d$?

Question

Suppose $a,b,c,d$ are consecutive prime numbers with $a<b<c<d$

Can $$a^b+b^c+c^d$$ be a prime number ?

On the one hand, I did not find a prime for $a\le 4723$ (maybe, someone double-checks this ?) , on the other hand, there are quartupels such that $a^b+b^c+c^d$ has no small prime factor. For example, the smallest prime factor of $$23^{29}+29^{31}+31^{37}$$ is $$1937815389893$$ having $13$ digits, the cofactor having $43$ digits, is prime as well. So, there is no easy prove that there is no such prime either.

Answer 1

The $n$-th number of the form you consider is of order $(n \log n)^{n \log n}= e^{n (\log n)^2 \log \log n}$.

The 'probability' that such a number is prime is $(n (\log n)^2 \log \log n)^{-1}$. The series $\sum (n (\log n)^2 \log \log n)^{-1}$ converges.

Thus, heuristically one expects only finitely many solutions. Since you did not yet find any among smaller numbers, chances are there are none.

I would guess to prove it is beyond current technology.

Given that there are examples without small factors it is also unlikely that a congruence condition can be found that excludes such numbers can exist.

Answer 2

Not an answer, but maybe useful for people who want to join in the hunt for a prime of the desired form :

If $p_n$ denotes the $n$-th prime and $$\large f_n:=p_n^{p_{n+1}}+p_{n+1}^{p_{n+2}}+p_{n+2}^{p_{n+3}}$$ then a number of the desired form can be just written as $f(n)$, where $n$ is a positive integer. For example, $f_5=11^{13}+13^{17}+17^{19}$. With this notation, the following array contains the postive numbers $n$ upto $3000$ , such that $f_n$ has no small prime factor :

[9, 16, 38, 41, 60, 71, 95, 97, 138, 154, 161, 165, 170, 212, 215, 223, 235, 
255, 275, 278, 306, 316, 329, 330, 335, 343, 346, 350, 353, 359, 361, 369, 
372, 377, 402, 406, 409, 434, 462, 466, 472, 476, 487, 490, 491, 494, 504, 
510, 535, 539, 554, 586, 592, 623, 628, 636, 637, 640, 654, 673, 677, 693, 
705, 718, 747, 768, 785, 827, 838, 842, 858, 868, 906, 912, 916, 918, 934, 
947, 954, 959, 960, 975, 985, 1026, 1028, 1034, 1042, 1059, 1061, 1063, 
1084, 1103, 1124, 1149, 1150, 1157, 1182, 1197, 1201, 1204, 1216, 1221, 
1240, 1248, 1273, 1290, 1297, 1301, 1302, 1309, 1310, 1335, 1341, 1350, 
1369, 1371, 1373, 1385, 1393, 1401, 1403, 1408, 1415, 1416, 1456, 1458, 
1459, 1474, 1483, 1488, 1494, 1497, 1505, 1516, 1521, 1524, 1526, 1557, 
1599, 1625, 1668, 1672, 1682, 1685, 1693, 1708, 1717, 1721, 1734, 1748,  
1761, 1787, 1798, 1814, 1848, 1870, 1875, 1916, 1919, 1923, 1931, 1955,    
1972, 1979, 1998, 2011, 2012, 2015, 2026, 2036, 2045, 2054, 2055, 2064, 
2065, 2078, 2080, 2084, 2099, 2123, 2150, 2155, 2184, 2192, 2197, 2207, 
2252, 2261, 2300, 2304, 2325, 2326, 2358, 2372, 2373, 2387, 2398, 2399, 
2411, 2425, 2427, 2438, 2449, 2464, 2492, 2495, 2519, 2526, 2534, 2540, 
2543, 2545, 2556, 2560, 2576, 2595, 2597, 2605, 2637, 2695, 2697, 2751, 
2765, 2769, 2770, 2784, 2806, 2832, 2842, 2855, 2871, 2902, 2919, 2922, 
2947, 2964, 2981, 2990]