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Old 2020-07-14, 12:48   #1
rogue
 
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Default Reservations fop x^y+y^x

This thread is to capture reservations and completed ranges per this page. There is another search for primes of this form, but that search is by decimal length not by range of x and y. Those reservations are not managed by this thread.

At this time all y have been tested for all x <= 13000. Note that y < x as we want x^y > y^x.
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Old 2020-07-14, 12:51   #2
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I will reserve all y for 13001 <= x <= 15000. I will also double-check from x=12501 to x = 13000.

I will do all I can to avoid stepping on the toes of the other search for primes of this form less than 100,000 decimal digits. If anything I will be double-checking their work. They will likely complete their search before I start testing the range they are working on.

Last fiddled with by rogue on 2020-07-14 at 12:56
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Old 2020-07-14, 17:59   #3
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The largest x^y+y^x occurs for y=x which for a given x-range occurs for the largest x in that range. Thus, the largest Leyland number up to x=13000 is 2*13000^13000, which has 53482 decimal digits. Here is a table of Leyland number decimal digits for largest x from 13000 to 30000 at intervals of 1000:

13000 53482
14000 58047
15000 62642
16000 67267
17000 71918
18000 76596
19000 81297
20000 86021
21000 90767
22000 95534
23000 100321
24000 105126
25000 109949
26000 114790
27000 119648
28000 124521
29000 129410
30000 134314

This will give you an idea of where the overlap between the two systems lies. Having checked all Leyland numbers smaller than (currently) 84734 decimal digits implies (barring errors) that I have checked all x smaller than 19728. That allows me to suggest that this table of x, y values (based on Andrey Kulsha's ordering) is complete. Of course it would be nice to have verification.
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Old 2020-07-14, 18:41   #4
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Quote:
Originally Posted by pxp View Post
The largest x^y+y^x occurs for y=x which for a given x-range occurs for the largest x in that range. Thus, the largest Leyland number up to x=13000 is 2*13000^13000, which has 53482 decimal digits. Here is a table of Leyland number decimal digits for largest x from 13000 to 30000 at intervals of 1000:

13000 53482
14000 58047
15000 62642
16000 67267
17000 71918
18000 76596
19000 81297
20000 86021
21000 90767
22000 95534
23000 100321
24000 105126
25000 109949
26000 114790
27000 119648
28000 124521
29000 129410
30000 134314

This will give you an idea of where the overlap between the two systems lies. Having checked all Leyland numbers smaller than (currently) 84734 decimal digits implies (barring errors) that I have checked all x smaller than 19728. That allows me to suggest that this table of x, y values (based on Andrey Kulsha's ordering) is complete. Of course it would be nice to have verification.
Thanks. In the worst case scenario I will be double-checking your work, which shouldn't hurt anyone. I don't expect that to take too long after sieving.

If double-checking reveals no missed primes, then I might forego double-checking for larger x.
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Old 2020-08-06, 15:16   #5
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I've decided to take the doublecheck to x=30000. This will cover y<x for all of t those x. This could be about 6 months of work, but I don't know for certain yet because I need to do a lot of sieving.

To help me with that, I have made changes to xyyxsieve (code is committed, but exe is not on sourceforge yet) to reduce the memory usage of the program. In the previous version, a range of 1000 x can take 8 GB of memory (ouch). The changes have reduced that memory requirement by a factor of 10. Another good result of that change is a boost in speed by about 30%.
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