20201220, 09:14  #1 
Nov 2016
101100000100_{2} Posts 
x^y*y^x+1 (Generalized HyperCulllen/Woodall) primes
Are there any test limit of y, for x=6 (6^y*y^6+1), x=10 (10^y*y^10+1), and x=13 (13^y*y^13+1)? There are no known primes for x=13, and the only known primes for x=6 and x=10 are 6^1*1^6+1 and 10^1*1^10+1

20201222, 01:14  #2  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
22164_{8} Posts 
Quote:
Also you must have a restriction of y>1, because for any prime p, x=p1, y=1 gives you a prime (so x={6,10}, y=1 are trivial solutions; there are infinite easy large ones. take x=M_{51}1, y=1 and you have yourself a huge x^y*y^x+1 prime). There are tons of primes, but they are all "small". I consider all of these small (but with x>20,000): 77387^6*6^773871 77098^13*13^77098+1 70106^6*6^70106+1 69870^23*23^69870+1 <95256> 64002^10*10^64002+1 57776^10*10^57776+1 45225^56*56^452251 39188^21*21^391881 39014^10*10^39014+1 33455^6*6^334551 33424^10*10^33424+1 26444^21*21^264441 25570^6*6^25570+1 23254^10*10^23254+1 23181^10*10^231811 20338^21*21^203381 Quote:
Obviously, you should not be searching for them in PRPtop, ...and if you search properly in UTM there are a few: Code:
     rank description digits who year      53471 10379^4062*4062^10379+1 53769 p90 2002 56329 8380^4057*4057^8380+1 46154 p90 2002 59130 6195^4022*4022^6195+1 37582 p90 2002 60036 5230^4032*4032^5230+1 33850 p90 2002 61977 4414^4000*4000^4414+1 30479 p90 2002 62221 49518^4*4^49518+1 29832 p14 2001      47287 18282^4013*4013^182821 82983 p90 2002 48199 16072^4077*4077^160721 75174 p90 2002 49310 14292^4049*4049^142921 68381 p90 2002 49347 14287^4002*4002^142871 68094 p90 2002 52539 11098^4011*4011^110981 56215 p90 2002 57722 5929^5010*5010^59291 40839 p90 2002 58989 6352^4041*4041^63521 38276 p90 2002 59165 5071^5004*5004^50711 37300 p90 2002 59223 5011^5000*5000^50111 37036 p90 2002 60219 5045^4058*4058^50451 33231 p90 2002      

20201222, 02:18  #3 
Jun 2003
1,579 Posts 

20201222, 02:25  #4  
Feb 2017
Nowhere
43×101 Posts 
Quote:
A nearly mindless numerical sweep with PariGP turned up the following results. I say "nearly" mindless because for x = 11 and 13, I did exclude odd values of y. Code:
? for(y=1,2000,n=6^y*y^6;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n1),print(y" "))) 1 + 1  4 + 16 + 17  20 + 46 + 94 + 97  1427  1975  ? for(y=1,2000,n=10^y*y^10;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n1),print(y" "))) 1 + 4 + 16 + 28 + 267  576 + 759 + ? forstep(y=2,2000,2,n=11^y*y^11;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n1),print(y" "))) 20  122  1040  1218 + ? forstep(y=2,2000,2,n=13^y*y^13;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n1),print(y" "))) 32  72  632  

20201222, 02:55  #5  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2474_{16} Posts 
Quote:


20201222, 14:12  #6  
Feb 2017
Nowhere
43×101 Posts 
Quote:
I just ran a simpleminded script that I reckoned would take seconds or minutes rather than hours or longer, in hopes it might turn up some likely suspects. My hopes were largely realized. I note that, according to the posted source, there are no primes of the form 13^y*y^13 + 1 for y up to 25000. I was unable to discern any obvious reason there couldn't be any primes of this form, and the limit 25000 is IME far too small to give the notion any serious credence. The forms are not sufficiently interesting to me to merit any serious effort on my part. I did notice that, if gcd(x,y) has any odd prime factor, then both x^y*y*x + 1 and x^y*y^x  1 have algebraic factorizations. If x and y are both even, the x^y*y^x  1 is a difference of two squares. Beyond that, excluding specific prime factors seemed to be routine congruencechasing. 

20201222, 21:15  #7  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2474_{16} Posts 
Quote:
... and one for "23+" : 69870^23*23^69870+1 ... and one for "53+" : 53^66934*66934^53+1 <115669 digits> 

20201222, 21:37  #8 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,333 Posts 
If we parse the old results carefully, there are actually many values > 13 without known primes.
GHC: 7 GHW: And interesting quasi"Riesel number" (there are no primes) for GHWs are 4, 16 (and every even square). However, "4" was "extensively tested" :) up to [100000], according to the table. Same is true, of course, for 27 on both sides (sum/diff of cubes). 
20201223, 01:56  #9 
Feb 2017
Nowhere
43·101 Posts 
I note the following analogs to corresponding results for Leyland numbers: Let N = x^y*y^x. If x and y are both odd, N is odd, so N  1 and N +1 are both even. If N > 3, both are composite. Of x and y are both even, N is a perfect square, so N  1 has algebraic factors. If gcd(x, y) has an odd prime factor, N  1 and N + 1 both have algebraic factors. If (say) x = (n*k)^n (n > 1 odd) then x^y and y^x are nth powers, so N = A^n for some A, whence N + 1 and N  1 have algebraic factors; if x = 4*k^4 and y is odd, then N = 4*A^4 for some A, and again N + 1 and N  1 both have algebraic factors. In particular, 4^y* y^4 + 1 is composite if y is odd and y > 1. Though of (limited) theoretical interest, the (n*k)^n and 4*k^4 are pretty thin on the ground. The even squares are much more plentiful. 
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