Primes n^n+2(n∈2k+1, k∈Z)

[Paimon2005]

I tried to found primes which can be expressed as n^n+2(n∈2k+1, k∈Z), and I ran a python code

import math
k=3
def findgreatestsqrt(num):
    low=1
    high=num
    while low<high:
        mid=(low+high)//2
        if mid*mid==num:
            low=mid
            break
        elif mid*mid<num:
            low=mid+1
        else:
            high=mid-1
    return low
def isprime(num,extrainfo=False):
    fac=1
    if num<2:
        return False
    if num==2 or num==3:
        return True
    for i in range(2,findgreatestsqrt(num)+1):
        if num%i==0:
            fac=i
            if extrainfo:
                print(str(num)+" can be divided by "+str(i))
            break
    if fac==1:
        return True
    else:
        return False
while True:
    resu=isprime(k**k+2,True)
    if resu==True:
        print(str(k)+"^"+str(k)+"+2 is a prime")
    else:
        print(str(k)+"^"+str(k)+"+2 is not a prime")
    k+=2

And I got some result

3^3+2 is a prime
3127 can be divided by 53
5^5+2 is not a prime
823545 can be divided by 3
7^7+2 is not a prime
387420491 can be divided by 59
9^9+2 is not a prime
285311670613 can be divided by 97
11^11+2 is not a prime
302875106592255 can be divided by 3
13^13+2 is not a prime
437893890380859377 can be divided by 23
15^15+2 is not a prime
827240261886336764179 can be divided by 7
17^17+2 is not a prime
1978419655660313589123981 can be divided by 3
19^19+2 is not a prime
5842587018385982521381124423 can be divided by 31
21^21+2 is not a prime
20880467999847912034355032910569 can be divided by 19
23^23+2 is not a prime
88817841970012523233890533447265627 can be divided by 3
25^25+2 is not a prime
443426488243037769948249630619149892805 can be divided by 5
27^27+2 is not a prime
2567686153161211134561828214731016126483471 can be divided by 51131
29^29+2 is not a prime
17069174130723235958610643029059314756044734433 can be divided by 3
31^31+2 is not a prime
129110040087761027839616029934664535539337183380515 can be divided by 5
33^33+2 is not a prime
1102507499354148695951786433413508348166942596435546877 can be divided by 2003
35^35+2 is not a prime
10555134955777783414078330085995832946127396083370199442519 can be divided by 3
37^37+2 is not a prime
112595147462071192539789448988889059930192105219196517009951961 can be divided by 229
39^39+2 is not a prime
1330877630632711998713399240963346255985889330161650994325137953643 can be divided by 139
41^41+2 is not a prime
17343773367030267519903781288812032158308062539012091953077767198995509 can be divided by 3
43^43+2 is not a prime

But when my computer tried to check 45^45+2, there was not any other result, I turned to a website, Numberempire( https://www.numberempire.com/primenu...check&_p1=2327 )
Numberempire told me that Number 248063644451341145494649182395412689744530581492654164321720600128173828127 (45^45+2) is not a prime
Then, I have tried more number(used another website, Wolframalpha( https://www.wolframalpha.com/input?i...+k%3D22+to+300 ), what I expected, I was told that there is not any prime among n^n+2(n∈2k+1, k∈Z, k∈(22,300))
I don't know that whether is 2nd prime among n^n+2(n∈2k+1, k∈Z) exists or not

[kar_bon]

See A100407, but the PRP for n=1349 is already proven prime here. The prime for n=737 is here.

Try using (Win)PFGW with a script like

ABC2 $a^$a+2
a: from 1 to 100000

to check faster and higher values.

[sweety439]

Originally Posted by Paimon2005

I tried to found primes which can be expressed as n^n+2(n∈2k+1, k∈Z)

Try this:

http://factordb.com/index.php?query=...e=200&format=1

Also, n should not be == 1 mod 3, as the number will be divisible by 3

There should be infinitely many primes of the form n^n+2, as there are no covering congruence, algebraic factorization, or combine of them for n^n+2, the dual of it is 2*n^n+1, there should be also infinitely many primes of this form, however, n^(n+2)+1 seems to be composite for all n > 30

[Paimon2005]

Originally Posted by kar_bon

See A100407, but the PRP for n=1349 is already proven prime here. The prime for n=737 is here.

Try using (Win)PFGW with a script like

ABC2 $a^$a+2
a: from 1 to 100000

to check faster and higher values.

ABC2 (2*$a+1)^(2*$a+1)+2
a: from 1 to 100000

I found 2 3-PRP
(2*368+1)^(2*368+1)+2
(2*674+1)^(2*674+1)+2

[sweety439]

Originally Posted by Paimon2005

ABC2 (2*$a+1)^(2*$a+1)+2
a: from 1 to 100000

I found 2 3-PRP
(2*368+1)^(2*368+1)+2
(2*674+1)^(2*674+1)+2

These two primes are already known, see https://oeis.org/A100407

You can try to find the smallest k >= 1 such that (2*n+1)^k+2 is prime, for n not divisible by 3

[kar_bon]

Originally Posted by sweety439

You can try to find the smallest k >= 1 such that (2*n+1)^k+2 is prime, for n not divisible by 3

There're countless of them, but the smallest is k=1 & n=1 -> 5.

[kar_bon]

Originally Posted by Paimon2005

ABC2 (2*$a+1)^(2*$a+1)+2
a: from 1 to 100000

Then take (OEIS says n has to be greater than 50000):

ABC2 $a^$a+2
a: from 50001 to 99999 step 2

[JeppeSN]

Originally Posted by sweety439

however, n^(n+2)+1 seems to be composite for all n > 30

For these, the exponent must be a power of two, so you would check:

62^64+1; 126^128+1; 254^256+1; 510^512+1; ...

The first many of them are already proven composite by Generalized Fermat prime searches. It seems extremely likely you are correct there are no (more) primes.

Similar things happen with n^n+1 and n^(n-2)+1.

/JeppeSN