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-   -   Leyland Primes (x^y+y^x primes) (https://www.mersenneforum.org/showthread.php?t=19347)

 pxp 2021-03-11 16:31

I'm guessing that I have used pfgw64 on some 5 million Leyland numbers since I started using it back in early July of last year. This is the first error encountered (this morning) using it:

Expr = 34048^5655+1*5655^34048
Detected in MAXERR>0.45 (round off check) in prp_using_gwnum
Iteration: 197019/424418 ERROR: ROUND OFF 0.5>0.45
PFGW will automatically rerun the test with -a1

 rogue 2021-03-11 18:22

[QUOTE=pxp;573421]I'm guessing that I have used pfgw64 on some 5 million Leyland numbers since I started using it back in early July of last year. This is the first error encountered (this morning) using it:

Expr = 34048^5655+1*5655^34048
Detected in MAXERR>0.45 (round off check) in prp_using_gwnum
Iteration: 197019/424418 ERROR: ROUND OFF 0.5>0.45
PFGW will automatically rerun the test with -a1[/QUOTE]

That does happen, but is rare. Fortunately it tried with a different FFT size automatically.

 pxp 2021-03-18 19:39

Leyland primes curve fit

1 Attachment(s)
I was curious about how many more new primes I was going to find in my current interval (#19) as well as the two subsequent ones (#20 & #22) so I decided to do a more formal calculation instead of my usual ballpark estimates. I first used the approach [URL="http://gladhoboexpress.blogspot.com/2015/05/indexing-leyland-primes.html"]back in 2015[/URL] to calculate a best fit curve (y = Leyland number index = ax^b) for the then 954 Leyland prime indices that I believed were sequential and used that curve to decide that the prime index of L(328574,15) — still the largest known Leyland prime — would be ~5550.

I used the 2222 Leyland prime indices that I currently have as sequential to recalculate the best fit. In the attached, that curve is red, contrasted with a green curve for the 2015 calculation. The green curve actually holds up pretty well until we get to ~1800. The recalculated L(328574,15) now comes in at index ~5908. But I wanted to know how many new primes I was going to find in the next couple of months. For interval #19, the suggested total will be ~88 (I have 80 as I write with another week or so to go). Interval #20 will yield ~90 and #22, ~97.

 pxp 2021-03-27 14:38

[QUOTE=pxp;572777]That makes L(48694,317) #2221.[/QUOTE]

I have examined all Leyland numbers in the seven gaps between L(48694,317) <121787>, #2221, and L(44541,746) <127955> and found 111 new primes. That makes L(44541,746) #2339.

So much for my March 18th calculated prediction (for this interval) of only 88 new primes. I do update a [URL="http://chesswanks.com/num/a094133.html"]sortable-columns version[/URL] of my [URL="http://chesswanks.com/num/a094133.txt"]Leyland primes indexing page[/URL] when I finish an interval or find a prime with a [I]y[/I] smaller than 1000. But it's too much effort to update it every time I find a new prime as I have to make three corrections to the html after each page conversion.

 pxp 2021-04-30 00:51

[QUOTE=pxp;574615]That makes L(44541,746) #2339.[/QUOTE]

I have examined all Leyland numbers in the four gaps between L(44541,746) <127955>, #2339, and L(49205,532) <134129> and found 99 new primes. That makes L(49205,532) #2442 and advances the index to L(49413,580), #2485.

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