Leyland Primes (x^y+y^x primes)

[pxp]

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]

Originally Posted by pxp

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

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

[pxp]

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 back in 2015 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]

Originally Posted by pxp

That makes L(48694,317) #2221.

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 sortable-columns version of my Leyland primes indexing page when I finish an interval or find a prime with a y 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]

Originally Posted by pxp

That makes L(44541,746) #2339.

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.

[pxp]

As my search of interval #22 winds down (ten or so day to go), I started (yesterday) the interval from L(299999,10) to L(300999,10). A preliminary estimate suggests that this will require some two-and-a-half months.

[NorbSchneider]

Another new PRP:
45^104608+104608^45, 172940 digits.

[pxp]

Originally Posted by pxp

That makes L(49205,532) #2442 and advances the index to L(49413,580), #2485.

I have examined all Leyland numbers in the two gaps between L(49413,580) <136550>, #2485, and L(49878,755) <143547> and found 123 new primes. That makes L(49878,755) #2610 and advances the index to L(45728,1905), #2691.

I believe that we have now all Leyland primes/PRPs < 150000 decimal digits or, equivalently, all prime/PRP L(x,y), x < 33180.

[bur]

Impressive compilation. Do you have data on which of the numbers are just PRPs? It would make a nice list of candidates for primo.

[kar_bon]

You can look at this table for a list of unproven numbers. I've not looked at those for a longer time, so some are verified and a certificate is available at FactorDB.
Just updated only 3 numbers, see the recent changes.
Dates and program taken from FDB.

[bur]

Ok, thanks.