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

[pxp]

Originally Posted by pxp

I will likely do a search for Leyland primes between L(49878,755) and L(149999,10), so as to establish a new beachhead for my advance.

I have added indicators for intervals #19-#21 to my list. I have also decided that the above mentioned "beachhead" is far too ambitious. My intended beachhead is now interval #21. I am still in the process of verifying my new list of Leyland numbers which runs from L(102999,10) to L(149999,10), 337553864 terms. A worthwhile guide is that for d > 11, L(d-1,10) is (likely) the smallest (base ten) d-digit term. The sorted-by-magnitude list will allow me to directly look up the Leyland number index of any (x,y) term in that range. It will also, of course, provide the seed (x,y) pairs needed to generate the ABC files for my intervals.

[NorbSchneider]

Another new PRP:
1678^28479+28479^1678, 91839 digits.

[pxp]

Originally Posted by pxp

That makes L(91382,9) #1660.

I have examined all Leyland numbers in the six gaps between L(91382,9) <87201>, #1660, and L(35829,302) <88857> and found 27 new primes. That makes L(35829,302) #1693.

[pxp]

Originally Posted by pxp

That makes L(35829,302) #1693.

I have examined all Leyland numbers in the two gaps between L(35829,302) <88857>, #1693, and L(37738,243) <90029> and found 20 new primes. That makes L(37738,243) #1715.

[pxp]

Originally Posted by pxp

A worthwhile guide is that for d > 11, L(d-1,10) is (likely) the smallest (base ten) d-digit term.

I began to wonder if any of these L(x,10) is prime. I'm doing a run on a list that I didn't sieve particularly deeply and I can say that for x < 300000 the answer is none. Perhaps this had already been determined. Of the prime L(x,y) that have x > 50000, I count 13 (current) solutions: (57285,2), (58046,9), (63880,3), (78296,3), (91382,9), (99069,2), (104824,5), (125330,3), (222748,3), (234178,9), (255426,11), (314738,9), (328574,15). All small y, which makes me wonder how far the discoverers allowed x to go (and for which y).

I've put a compilation of small-y solutions (y <= 1000) here.

[pxp]

Originally Posted by pxp

That makes L(37738,243) #1715.

I have examined all Leyland numbers in the two gaps between L(37738,243) <90029>, #1715, and L(38030,249) <91128> and found 17 new primes. That makes L(38030,249) #1734 and advances the index to L(37614,265), #1735.

[pxp]

Originally Posted by pxp

My intended beachhead is now interval #21.

I decided to take my sieving of interval #21 to 1e10 and that still has a couple of days to go. In the meantime I am pfgw-ing recently assigned (and already sieved) interval #28 [L(148999,10) - L(149999,10)] and have now my first hit therein:

33845^26604+26604^33845 is 3-PRP!

I'm not sure factordb.com will PRP this for me. I noticed that Norbert's PRPTop submissions for a couple of his larger Leyland primes has a list of prime-PRPs from prime 2 to 11. Which brings me to ask why pfgw default reports only 3-PRPs. How does one get it to do other primes? Is it even necessary?

[rogue]

Originally Posted by pxp

I decided to take my sieving of interval #21 to 1e10 and that still has a couple of days to go. In the meantime I am pfgw-ing recently assigned (and already sieved) interval #28 [L(148999,10) - L(149999,10)] and have now my first hit therein:

33845^26604+26604^33845 is 3-PRP!

I'm not sure factordb.com will PRP this for me. I noticed that Norbert's PRPTop submissions for a couple of his larger Leyland primes has a list of prime-PRPs from prime 2 to 11. Which brings me to ask why pfgw default reports only 3-PRPs. How does one get it to do other primes? Is it even necessary?

Use -b to choose a different base for the PRP test.

[pxp]

Originally Posted by pxp

That makes L(38030,249) #1734 and advances the index to L(37614,265), #1735.

I have examined all Leyland numbers in the ten gaps between L(37614,265) <91148>, #1735, and L(40210,287) <98832> and found 117 new primes. That makes L(40210,287) #1862 and advances the index to L(40945,328), #1930.

That completes interval #14 which I did in two parts. The second (larger Leyland numbers) part, which I did first, ended up with 69 PRPs. Because the first (smaller Leyland numbers) part started off with roughly an identical quantity (~21919300) of Leyland numbers as the second part, I was expecting at least 69 PRPs in it as well, but it ended up with only 57 PRPs.

I'm well on my way to completing (likely by October 12th) intervals #15, #16, and #28.

[rogue]

I have written a small program that converts pxp's text list of x^x+y^x primes/prps into an html table that has sortable columns. The source and html that it generates is attached.

[pxp]

Originally Posted by rogue

I have written a small program that converts pxp's text list of x^x+y^x primes/prps into an html table that has sortable columns.

Thank you. Fortunately I had written a note to myself from the last time I ran a .cpp program.

g++ xyyx.cpp -o xyyx

xyyx.cpp:60:56: warning: format specifies type 'unsigned long long *' but the argument has type 'long *'
      [-Wformat]
         if (sscanf(ptr, "%u %llu %u (%u,%u)", &index, &leylandNumber, &length, &x, &y) != 5)

The %llu and &leylandNumber were underlined. This was followed by a very similar warning ending in !=4 and finally a third one relating to an fprintf item containing the two offending variables. In spite of the warnings the created xyyx ran to create a list.html from a list.txt.

I can probably run this every time I update my a094133.txt document and share it here. A couple of minor issues: Christ van Willegen and Jens Kruse Andersen have lost their surnames and Göran Hemdal has lost the umlauted o (I assume that it is visible in the .txt version).