• #### (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
* Origin: fsxNet Usenet Gateway (21:1/5)
• 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

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• 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
--
We are no longer hunters and nomads. No longer awed and frightened, as we have gained some understanding of the world in which we live. As such, we can cast aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
The text passage contains a mathematical question about when the expression \(a^b + b^a\) is a prime number, and it provides several examples of pairs \((a, b)\) for which the expression is prime. It also includes some observations and explanations about the patterns and properties of these pairs.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
• #### (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
* Origin: fsxNet Usenet Gateway (21:1/5)
• 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

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• 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
--
We are no longer hunters and nomads. No longer awed and frightened, as we have gained some understanding of the world in which we live. As such, we can cast aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
user
The text passage contains a mathematical question about when the expression \(a^b + b^a\) is a prime number, and it provides several examples of pairs \((a, b)\) for which the expression is prime. It also includes some observations and explanations about the patterns and properties of these pairs.

TYPE: SINGLE_QUESTION_WITH_ANSWER