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

[pxp]

Originally Posted by pxp

The odds of finding a d-digit Leyland prime where d ~ 100000 are (empirically) about 1 in 75. In other words, one should expect to find about 40 Leyland primes with digit-size ranging from 100000 to 103000. What are the odds when d ~ 400000 ?

My estimate is about 1 in 100. So about 30 Leyland primes with digit-size ranging from 400000 to 403000. Seems like a lot.

[pxp]

Norbert Schneider recently PRP'd L(47012,297). At 116250 decimal digits, this is now the 6th largest-known Leyland prime. Moreover, this gives Norbert five of the top ten. Congratulations!

[NorbSchneider]

I search in the interval 13,000<=x<=15,000, for new PRPs
and also doublecheck the known PRPs.
Currenttly I reached x=13,800, so far no new PRPs and the
known PRPs are confirmed.

Hans, what is your next range after the 100,000 digits is finished?

[pxp]

Originally Posted by pxp

The odds of finding a d-digit Leyland prime where d ~ 100000 are (empirically) about 1 in 75. In other words, one should expect to find about 40 Leyland primes with digit-size ranging from 100000 to 103000.

I have 51.

Beginning on 26 June 2017, I am today done with my search of Leyland numbers between L(40210,287) <98832> and L(40945,328) <103013>. I found 67 new primes. The (x,y) values of the 25046458 Leyland numbers in the gap were precomputed, sorted by size, and packaged into bundles of 88670. Each bundle was assigned to an available core on one of my Macs.

The computations were done in Mathematica (versions 8 or 9) where each Leyland number was checked for GCD(x,y)==1 before applying PrimeQ. A given bundle would take from four to six weeks to check. There's no point in listing the new primes. Refer to my Leyland prime indexing effort which is always up-to-date.

Norbert asked what I'm doing next. I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years. I've already started.

[rogue]

Originally Posted by pxp

I have 51.

Beginning on 26 June 2017, I am today done with my search of Leyland numbers between L(40210,287) <98832> and L(40945,328) <103013>. I found 67 new primes. The (x,y) values of the 25046458 Leyland numbers in the gap were precomputed, sorted by size, and packaged into bundles of 88670. Each bundle was assigned to an available core on one of my Macs.

The computations were done in Mathematica (versions 8 or 9) where each Leyland number was checked for GCD(x,y)==1 before applying PrimeQ. A given bundle would take from four to six weeks to check. There's no point in listing the new primes. Refer to my Leyland prime indexing effort which is always up-to-date.

Norbert asked what I'm doing next. I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years. I've already started.

I'm confused. Did you do the PRP testing with Mathematica?

[pxp]

Yes. PrimeQ is Mathematica's PRP test.

[a1call]

I looked this up. Others might find it useful. The subject has come up more than once in the past.

The Rabin-Miller strong pseudoprime test is a particularly efficient test. The Wolfram Language implements the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test as the primality test used by the function PrimeQ[n]. Like many such algorithms, it is a probabilistic test using pseudoprimes. In order to guarantee primality, a much slower deterministic algorithm must be used. However, no numbers are actually known that pass advanced probabilistic tests (such as Rabin-Miller) yet are actually composite.

http://mathworld.wolfram.com/PrimalityTest.html

[pxp]

Originally Posted by pxp

I'm back to extending the prime indices which have languished at L(17691,1508) <56230>, #1179, for these almost-two years.

I just looked it up. The index reached #1179 on 23 Aug 2016 already!

[pxp]

Originally Posted by pxp

The index reached #1179 on 23 Aug 2016 already!

I have examined all Leyland numbers in the gap between L(17691,1508) <56230>, #1179, and L(17605,1908) <57755> and found 19 new primes. That makes L(17605,1908) #1199.

[pxp]

Originally Posted by pxp

That makes L(17605,1908) #1199.

I have examined all Leyland numbers in the three gaps between L(17605,1908) <57755>, #1199, and L(26530,163) <58690> and found 12 new primes. That makes L(26530,163) #1214.

[pxp]

Originally Posted by pxp

That makes L(26530,163) #1214.

I have examined all Leyland numbers in the gap between L(26530,163) <58690>, #1214, and L(125330,3) <59798> and found 8 new primes. That makes L(125330,3) #1223 and advances the index to L(28468,129), #1226.