• #### (ABBA primes) -- When is (a^b + b^a) a prime number ?

From henhanna@gmail.com@21:1/5 to All on Sun Aug 21 20:11:13 2022
( 30 ^ 13 + 13 ^ 30 ) ---- Is NOT a prime number

When is (a^b + b^a) a prime number ?

------ pls (if you know the Answer already) pls wait 3+ days before posting answers or hints.

C:\Python> py abba.py

( 2 , 3 ) 8+9= 17 is prime

( 2 , 9 ) 593 is prime

( 2 , 15 ) 32993 is prime

( 2 , 21 ) 2097593 is prime

( 2 , 33 ) 8589935681 is prime

( 3 , 56 ) 523347633027360537213687137 is prime

( 5 , 24 ) 59604644783353249 is prime

( 7 , 54 ) 4318114567396436564035293097707729426477458833 is prime

( 8 , 69 ) 205688069665150755269371147819668813122841983204711281293004769 is prime

( 8 , 519 )

( 9 , 76 ) 3329896365316142756322307042065269797678257903507506764421250291562312417 is prime

( 9 , 122 ) 261568927457882874608733211757582315090892217214195250256575658313972901281170319830426649720495055337775965208077073 is prime

( 9 , 422 )

( 15 , 32 ) 43143988327398957279342419750374600193 is prime

( 20 , 357 ) ( 20 , 471 )

( 21 , 68 ) 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129 is prime

( 21 , 782 )

( 32 , 135 )
( 32 , 717 )

( 33 , 38 ) 5052785737795758503064406447721934417290878968063369478337 is prime

( 34 , 75 ) ( 34 , 773 )

( 36 , 185 )

( 45 , 158 )

( 51 , 206 )

( 54 , 983 )

( 56 , 87 ) ( 56 , 477 )

( 65 , 144 )

( 67 , 114 ) 14877416035581437625382418693025659213718389161995860818124841388673684963203665153674781821433446993366770573625979847557897428218464508224911011186563057321746523584348117445155146293741592207500868288335433 is prime

( 68 , 927 )

( 76 , 215 )

( 80 , 81 )

( 87 , 248 ) ( 87 , 734 )

( 91 , 318 ) ( 91 , 636 )

( 97 , 114 )

( 98 , 171 ) ( 98 , 435 ) ( 98 , 663 )

( 111 , 322 )

( 122 , 333 ) ( 133 , 160 ) ....................................

( 200 , 237 ) 22085588309729804119791218759286481447843548710945236976520077516157748090572339238804468275731523465416767006325035024374407725635084544633776118082536633700263560665600734113232016803228139257501752170351377101892736071367151701
362486456643554714347467014996286162525276048043752820824400823564508992712106990676891300357999356852449353413375180032277905651741204247785529059353863025688172375548002480181584577315416055399778283298923604389312686818276175269418013471830160500647812
5705120066279786312075737518812303625846500724858615981588001 is prime

--- SoupGate-Win32 v1.05
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• From henhanna@gmail.com@21:1/5 to henh...@gmail.com on Tue Aug 23 16:37:17 2022
(this list is not complete... but)

i'm noticing that ...

0. A,B are ( one even and the other odd )

1. After ( 7 , 54 ) (is ABBA prime), primes are rare

( 34 , 773 ) is an exception (773 is prime)

2. One of A, B tends to be a multiple of 3

again , ( 34 , 773 ) is an exception

On Sunday, August 21, 2022 at 8:11:14 PM UTC-7, henh...@gmail.com wrote:
( 30 ^ 13 + 13 ^ 30 ) ---- Is NOT a prime number

When is (a^b + b^a) a prime number ?

------ pls (if you know the Answer already) pls wait 3+ days before posting answers or hints.

C:\Python> py abba.py

( 2 , 3 ) 8+9= 17 is prime

( 2 , 9 ) 593 is prime

( 2 , 15 ) 32993 is prime

( 2 , 21 ) 2097593 is prime

( 2 , 33 ) 8589935681 is prime

( 3 , 56 ) 523347633027360537213687137 is prime

( 5 , 24 ) 59604644783353249 is prime

( 7 , 54 ) 4318114567396436564035293097707729426477458833 is prime

( 8 , 69 ) 205688069665150755269371147819668813122841983204711281293004769 is prime

( 8 , 519 )

( 9 , 76 ) 3329896365316142756322307042065269797678257903507506764421250291562312417 is prime

( 9 , 122 ) 261568927457882874608733211757582315090892217214195250256575658313972901281170319830426649720495055337775965208077073 is prime

( 9 , 422 )

( 15 , 32 ) 43143988327398957279342419750374600193 is prime

( 20 , 357 ) ( 20 , 471 )

( 21 , 68 ) 814539297859635326656252304265822609649892589675472598580095801187688932052096060144958129 is prime

( 21 , 782 )

( 32 , 135 )
( 32 , 717 )

( 33 , 38 ) 5052785737795758503064406447721934417290878968063369478337 is prime

( 34 , 75 ) ( 34 , 773 )

( 36 , 185 )

( 45 , 158 )

( 51 , 206 )

( 54 , 983 )

( 56 , 87 ) ( 56 , 477 )

( 65 , 144 )

( 67 , 114 ) 14877416035581437625382418693025659213718389161995860818124841388673684963203665153674781821433446993366770573625979847557897428218464508224911011186563057321746523584348117445155146293741592207500868288335433 is prime

( 68 , 927 )

( 76 , 215 )

( 80 , 81 )

( 87 , 248 ) ( 87 , 734 )

( 91 , 318 ) ( 91 , 636 )

( 97 , 114 )

( 98 , 171 ) ( 98 , 435 ) ( 98 , 663 )

( 111 , 322 )

( 122 , 333 ) ( 133 , 160 ) ....................................

( 200 , 237 ) 22085588309729804119791218759286481447843548710945236976520077516157748090572339238804468275731523465416767006325035024374407725635084544633776118082536633700263560665600734113232016803228139257501752170351377101892736071367151701362486456
643554714347467014996286162525276048043752820824400823564508992712106990676891300357999356852449353413375180032277905651741204247785529059353863025688172375548002480181584577315416055399778283298923604389312686818276175269418013471830160500647812570512006
6279786312075737518812303625846500724858615981588001 is prime

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• From Phil Carmody@21:1/5 to henh...@gmail.com on Sun Aug 28 12:39:25 2022
"henh...@gmail.com" <henhanna@gmail.com> writes:
(this list is not complete... but)

i'm noticing that ...

0. A,B are ( one even and the other odd )

Because odd^odd+odd^odd = even, and even^even+even^even = even.

1. After ( 7 , 54 ) (is ABBA prime), primes are rare

( 34 , 773 ) is an exception (773 is prime)

They grow quickly, so the primes thin out. Nothing unexpected.

2. One of A, B tends to be a multiple of 3

again , ( 34 , 773 ) is an exception

That ought to be explainable in terms of how often 3 will be a factor
of x^y+y^x, as x,y range over the different exponents, but a quick
check didn't show why that would select 3|xy specifically.

The only obvious divisibility pattern I can see is that the x and y
terms are less likely to be one less than an odd prime. This is because (p-1)^odd == -1 (mod p), and anything_coprime_to_p^(p-1) == 1 (mod p),
so (p-1)^odd + odd^(p-1) == 0 (mod p) unless p|odd.

Phil
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