x^y*y^x+-1 (Generalized HyperCulllen/Woodall) primes

[sweety439]

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

[Batalov]

Originally Posted by sweety439

...and the only known primes for x=6 and x=10 are 6^1*1^6+1 and 10^1*1^10+1

By that, Captain Obvious deduces that you think that y<x makes any sense for this symmetrical form. Except it doesn't.
Also you must have a restriction of y>1, because for any prime p, x=p-1, y=1 gives you a prime (so x={6,10}, y=1 are trivial solutions; there are infinite easy large ones. take x=M₅₁-1, 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*6^77387-1
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*56^45225-1
39188^21*21^39188-1
39014^10*10^39014+1
33455^6*6^33455-1
33424^10*10^33424+1
26444^21*21^26444-1
25570^6*6^25570+1
23254^10*10^23254+1
23181^10*10^23181-1
20338^21*21^20338-1

Originally Posted by sweety439

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

Because none of them are in factordb.com, no one likely searched for them anywhere deep enough.
Obviously, you should not be searching for them in PRPtop,
...and if you search properly in UTM there are a few:

-----  -------------------------------- ------- ----- ---- 
 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*4013^18282-1            82983   p90 2002 
48199  16072^4077*4077^16072-1            75174   p90 2002 
49310  14292^4049*4049^14292-1            68381   p90 2002 
49347  14287^4002*4002^14287-1            68094   p90 2002 
52539  11098^4011*4011^11098-1            56215   p90 2002 
57722  5929^5010*5010^5929-1              40839   p90 2002 
58989  6352^4041*4041^6352-1              38276   p90 2002 
59165  5071^5004*5004^5071-1              37300   p90 2002 
59223  5011^5000*5000^5011-1              37036   p90 2002 
60219  5045^4058*4058^5045-1              33231   p90 2002 
-----  -------------------------------- ------- ----- ----

[Citrix]

https://harvey563.tripod.com/

These are called Generalized HyperCulllen/Woodall.

[Dr Sardonicus]

Originally Posted by sweety439

<snip>
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

I suggest you upgrade your abacus.

A nearly mindless numerical sweep with Pari-GP turned up the following results. I say "nearly" mindless because for x = 11 and 13, I did exclude odd values of y.

? for(y=1,2000,n=6^y*y^6;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n-1),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(n-1),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(n-1),print(y" -")))
20 -
122 -
1040 -
1218 +

? forstep(y=2,2000,2,n=13^y*y^13;if(ispseudoprime(n+1),print(y" +"));if(ispseudoprime(n-1),print(y" -")))
32 -
72 -
632 -

[Batalov]

Originally Posted by Citrix

https://harvey563.tripod.com/

These are called Generalized HyperCulllen/Woodall.

The limits are really tiny there (for small x or "k").

[Dr Sardonicus]

Originally Posted by Citrix

https://harvey563.tripod.com/

These are called Generalized HyperCulllen/Woodall.

The limits may be "tiny" by today's standards, but are way beyond where I was willing even to try to look.

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 congruence-chasing.

[Batalov]

Originally Posted by Dr Sardonicus

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.

I got one for this form : 77098^13*13^77098+1
... and one for "23+" : 69870^23*23^69870+1
... and one for "53+" : 53^66934*66934^53+1 <115669 digits>

[Batalov]

If we parse the old results carefully, there are actually many values > 13 without known primes.
GHC: 7 13 19 23 27 31 47 53 ...
GHW: 16 21 22 23 27 29 31 36 45 49 50 51 52 55 56 ...

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).

[Dr Sardonicus]

Originally Posted by Batalov

I got one for this form
77098^13*13^77098+1

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 n-th 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.