Prime number expressible as $2^2+3^3+5^5+7^7+11^{11}+\cdots$

Solution 1:

The standard heuristic for such things is this.

Given a (growing) sequence $(a_n)_{n\in \mathbb N}$ where there is no apparent reason for a number in there to be prime or not consider (as the probability that a number of a certain rough size $N$ is prime is $(\log N)^{-1}$)

$$\sum_{n=1}^{\infty} \frac{1}{ \log a_n} $$

If this converges there should be only finitely many primes in your sequence.

Let us do this for you sequence. We have that $a_n \ge p_n^{p_n}$ and as $p_n \sim n \log n $ let us estimate $a_n$ as $(n \log n)^{n \log n}$ so $\log a_n$ is about $n (\log n)^2 $ (I drop all lower order terms).

The series $$\sum_{n =2}^{\infty} \frac{1}{ n (\log n)^2 } $$ converges, so we expect only finitely many primes in that sequence.

It thus is conceivable you found all of them, but there is likely no way to prove this with current "technology." Note that there is also no proof yet that there are only finitely many Fermat primes, which should be rather easier since that sequence grows faster and is generally easier to handle.

Solution 2:

If a prime number of the form $$2^2+3^3+5^5+7^7+...+p^p$$ with $p>89$ exists, $p$ must be greater than $8731$ and the number must have more than $34,000$ digits.

The candidates for $2^2+3^3+5^5+7^7+...+p^p$ (no prime factor $\le 104,729$) upto $p=10^5$ are :

[7, 43, 89, 107, 193, 229, 251, 397, 433, 683, 701, 971, 1013, 1109, 1193, 1223, 1231, 1277, 1459, 1493, 1511, 1531, 1871, 2143, 2161, 2213, 2543, 2551, 2617, 2851, 2861, 3023, 3541, 3923, 3989, 4093, 4447, 4597, 4783, 4813, 5003, 5227, 5333, 5351, 5437, 5563, 5591, 5779, 5839, 6173, 7103, 7253, 7541, 7549, 7741, 7757, 7873, 8353, 8543, 8699, 8731, 8861, 9323, 9397, 9413, 9439, 9613, 9623, 9631, 9743, 9781, 9871, 10061, 10139, 10333, 10429, 10729, 10847, 10867, 11383, 11941, 11959, 12119, 12239, 12277, 12497, 12619, 12653, 12923, 13033, 13723, 13759, 13933, 13967, 14221, 14249, 14341, 14461, 14557, 14779, 14851, 14887, 15149, 15319, 15401, 15443, 15727, 15739, 15803, 16087, 16097, 16189, 16447, 17093, 17383, 17419, 17551, 17737, 17909, 18899, 18919, 19219, 19373, 19417, 19441, 19553, 19583, 19813, 19919, 20143, 20663, 21407, 22391, 22543, 22691, 22871, 22921, 23561, 23567, 23603, 23789, 24007, 24077, 24103, 24137, 24203, 24359, 24373, 24421, 24473, 25153, 25981, 26729, 26839, 26927, 26987, 27059, 27277, 27487, 27803, 27851, 27947, 28001, 28499, 28579, 28753, 29303, 29587, 29741, 29759, 29789, 29837, 29881, 30011, 30097, 30319, 30689, 31393, 31849, 32089, 32257, 32503, 32971, 33113, 33181, 33349, 33391, 33427, 33521, 33773, 33997, 34141, 34667, 34693, 34721, 34847, 35257, 35419, 35603, 36299, 36709, 37087, 37879, 38371, 38603, 38707, 39343, 39373, 40427, 40763, 40787, 41149, 41201, 41257, 41593, 41897, 41911, 42089, 42463, 42557, 43037, 43067, 43451, 43661, 43691, 43717, 44101, 44179, 44537, 44819, 45197, 45337, 45757, 45823, 46261, 46399, 46933, 47093, 47221, 47293, 47629, 47659, 47699, 47981, 48299, 48371, 48479, 48767, 49117, 49537, 49633, 49853, 49993, 50101, 50273, 50321, 50753, 50789, 51287, 51511, 51871, 52553, 52691, 52883, 53161, 53239, 53309, 53327, 54037, 54139, 54311, 55103, 55147, 55171, 55439, 55987, 56369, 56383, 56629, 56783, 56809, 57331, 57559, 57923, 58013, 58129, 58337, 58379, 58427, 58679, 59359, 59669, 60103, 60127, 60169, 60649, 60887, 61483, 61861, 62233, 62347, 62473, 62653, 62861, 62939, 62983, 63067, 63197, 63439, 63671, 63737, 63853, 64067, 64171, 64237, 64579, 64661, 64763, 65089, 65147, 65407, 65449, 65951, 65983, 66643, 67121, 67169, 67531, 67763, 68281, 68881, 69557, 69763, 69991, 70451, 70501, 70573, 70627, 71209, 71237, 71633, 71707, 71741, 71821, 71879, 71887, 72341, 72937, 73091, 73459, 73643, 73973, 74027, 74149, 74203, 74363, 74653, 74891, 74941, 75017, 75109, 76597, 76847, 76919, 77317, 77339, 77369, 77699, 78301, 78583, 78607, 78781, 78877, 78929, 79193, 79241, 79393, 79433, 79609, 79769, 80147, 80513, 80629, 80761, 80963, 81083, 81409, 81569, 82037, 82301, 82339, 82393, 83117, 83311, 83341, 83621, 83701, 84503, 84967, 85009, 86413, 86693, 86851, 87013, 87071, 87151, 87277, 87629, 88007, 88321, 88379, 88547, 88609, 89209, 89393, 89431, 89477, 89561, 89597, 89627, 89939, 89983, 90397, 90709, 91457, 91631, 91961, 92033, 92179, 92233, 92363, 92801, 92927, 93047, 93133, 93887, 94063, 94307, 94793, 94873, 95561, 95621, 95701, 95957, 95971, 96149, 96167, 96323, 96461, 96589, 96667, 96697, 96823, 96847, 96979, 97021, 97073, 97649, 97871, 98347, 98429, 98621, 98993, 99347, 99377, 99719]

I used this PARI/GP-program for the search :

? s=0;p=1;while(p<10^5,p=nextprime(p+1);s=s+p^p;if(gcd(s,u)==1,print1(p," ");if(
ispseudoprime(s,1)==1,print;print("********************",p);print)))
7
********************7

43 89
********************89

107 193 229 251 397 433 683 701 971 1013 1109 1193 1223 1231 1277 1459 1493 1511
 1531 1871 2143 2161 2213 2543 2551 2617 2851 2861 3023 3541 3923 3989 4093 4447
 4597 4783 4813 5003 5227 5333 5351 5437 5563 5591 5779 5839 6173 7103 7253 7541
 7549 7741 7757 7873 8353 8543 8699 8731 8861 9323