• #### ( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

From henhanna@gmail.com@21:1/5 to All on Fri Aug 19 22:38:54 2022
------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days) before posting answers or hints.

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days) before posting answers or hints.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Aug 21 14:35:07 2022
On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049

sum = 2619995643649960903610551432833049

Trial and error shows that this is composite, equaling
31 * 84515988504837448503566175252679
= 31 * 59 * 1432474381437922855992647038181
(no other factors > 1 and < 100)

There's also
https://www.calculator.net/factor-calculator.html
but it only works on numbers up to 10^13, whereas we've only reduced
things to around 10^30 so far.

If you don't have a calculator handy that can handle numbers this large,
then you can also work it out by choosing a potential prime factor, then reducing intermediate computations modulo that prime, e.g.
30 ^ 1 = 30 (mod 31)
30 ^ 2 = 30 * 30 = 900 = 1 (mod 31)
30 ^ 3 = 1 * 30 = 30 (mod 31)
...
30 ^ 13 = 30 (mod 31)
and
13 ^ 1 = 13 (mod 31)
13 ^ 2 = 13 * 13 = 169 = 14 (mod 31)
13 ^ 3 = 14 * 13 = 182 = 27 (mod 31)
13 ^ 4 = 27 * 13 = 351 = 10 (mod 31)
13 ^ 5 = 10 * 13 = 130 = 6 (mod 31)
13 ^ 30 = (13 ^ 5) ^ 6 = 6 ^ 6 = 46656 = 1 (mod 31)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Gareth Taylor@21:1/5 to henh...@gmail.com on Sun Aug 21 23:41:21 2022
In article <tdu8eh\$1pod\$1@gioia.aioe.org>,
Edward Murphy <emurphy42@zoho.com> wrote:

On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049

sum = 2619995643649960903610551432833049

Like many of these recent maths questions we're getting, it falls to a
bit of basic number theory (probably found in most first-year maths
degrees, or maybe even earlier), in this case Fermat's Little Theorem.

This states that if p is prime and x isn't a multiple of p, then x^{p-1}
is congruent to 1 modulo p. Or, in less mathsy language, that p divides x^{p-1} - 1. So we work modulo 31.

Since 30 == -1 mod 31, we have 30^13 == (-1)^13 == -1 mod 31.

And by Fermat, we have 13^30 == 1 mod 31.

So their sum of 0 modulo 31, and so 31 divides 30^13 + 13^30.

Gareth

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Jonathan Dushoff@21:1/5 to Edward Murphy on Mon Aug 22 09:17:57 2022
Space:

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If you have bc, you probably have factor, which will tell you:

2619995643649960903610551432833049: 31 59 8543 59316821 2826822827838696527

On Sunday, August 21, 2022 at 5:35:17 PM UTC-4, Edward Murphy wrote:

On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049

Trial and error shows that this is composite, equaling
31 * 84515988504837448503566175252679
= 31 * 59 * 1432474381437922855992647038181
(no other factors > 1 and < 100)

There's also
https://www.calculator.net/factor-calculator.html
but it only works on numbers up to 10^13, whereas we've only reduced
things to around 10^30 so far.

If you don't have a calculator handy that can handle numbers this large,
then you can also work it out by choosing a potential prime factor, then reducing intermediate computations modulo that prime, e.g.
30 ^ 1 = 30 (mod 31)
30 ^ 2 = 30 * 30 = 900 = 1 (mod 31)
30 ^ 3 = 1 * 30 = 30 (mod 31)
...
30 ^ 13 = 30 (mod 31)
and
13 ^ 1 = 13 (mod 31)
13 ^ 2 = 13 * 13 = 169 = 14 (mod 31)
13 ^ 3 = 14 * 13 = 182 = 27 (mod 31)
13 ^ 4 = 27 * 13 = 351 = 10 (mod 31)
13 ^ 5 = 10 * 13 = 130 = 6 (mod 31)
13 ^ 30 = (13 ^ 5) ^ 6 = 6 ^ 6 = 46656 = 1 (mod 31)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Richard Heathfield@21:1/5 to Jonathan Dushoff on Mon Aug 22 18:45:15 2022
On 22/08/2022 5:17 pm, Jonathan Dushoff wrote:

Space:

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If you have bc, you probably have factor, which will tell you:

2619995643649960903610551432833049: 31 59 8543 59316821 2826822827838696527

So it's not /just/ a prime number; it's several!

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Ilan Mayer@21:1/5 to Edward Murphy on Tue Aug 23 10:53:26 2022
On Sunday, August 21, 2022 at 5:35:17 PM UTC-4, Edward Murphy wrote:
On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049

sum = 2619995643649960903610551432833049

Trial and error shows that this is composite, equaling
31 * 84515988504837448503566175252679
= 31 * 59 * 1432474381437922855992647038181
(no other factors > 1 and < 100)

2619995643649960903610551432833049 = 31 * 59 * 8543 * 59316821 * 2826822827838696527

There's also
https://www.calculator.net/factor-calculator.html
but it only works on numbers up to 10^13, whereas we've only reduced
things to around 10^30 so far.

If you don't have a calculator handy that can handle numbers this large,
then you can also work it out by choosing a potential prime factor, then reducing intermediate computations modulo that prime, e.g.
30 ^ 1 = 30 (mod 31)
30 ^ 2 = 30 * 30 = 900 = 1 (mod 31)
30 ^ 3 = 1 * 30 = 30 (mod 31)
...
30 ^ 13 = 30 (mod 31)
and
13 ^ 1 = 13 (mod 31)
13 ^ 2 = 13 * 13 = 169 = 14 (mod 31)
13 ^ 3 = 14 * 13 = 182 = 27 (mod 31)
13 ^ 4 = 27 * 13 = 351 = 10 (mod 31)
13 ^ 5 = 10 * 13 = 130 = 6 (mod 31)
13 ^ 30 = (13 ^ 5) ^ 6 = 6 ^ 6 = 46656 = 1 (mod 31)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Jonathan Dushoff on Tue Aug 23 11:17:32 2022
On Monday, August 22, 2022 at 9:17:59 AM UTC-7, Jonathan Dushoff wrote:

If you have bc, you probably have factor, which will tell you:

2619995643649960903610551432833049: 31 59 8543 59316821 2826822827838696527

i couldn't find Factor( ) or Factorize( ) in
http://sciencesoft.at/bc
http://sciencesoft.at/bc/?lang=en

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

( 30 ^ x + x ^ 30 ) ---- this a prime only when X=1

On Sunday, August 21, 2022 at 5:35:17 PM UTC-4, Edward Murphy wrote:

On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049
Trial and error shows that this is composite, equaling
31 * 84515988504837448503566175252679
= 31 * 59 * 1432474381437922855992647038181
(no other factors > 1 and < 100)

There's also
https://www.calculator.net/factor-calculator.html
but it only works on numbers up to 10^13, whereas we've only reduced
things to around 10^30 so far.

If you don't have a calculator handy that can handle numbers this large, then you can also work it out by choosing a potential prime factor, then reducing intermediate computations modulo that prime, e.g.
30 ^ 1 = 30 (mod 31)
30 ^ 2 = 30 * 30 = 900 = 1 (mod 31)
30 ^ 3 = 1 * 30 = 30 (mod 31)
...
30 ^ 13 = 30 (mod 31)
and
13 ^ 1 = 13 (mod 31)
13 ^ 2 = 13 * 13 = 169 = 14 (mod 31)
13 ^ 3 = 14 * 13 = 182 = 27 (mod 31)
13 ^ 4 = 27 * 13 = 351 = 10 (mod 31)
13 ^ 5 = 10 * 13 = 130 = 6 (mod 31)
13 ^ 30 = (13 ^ 5) ^ 6 = 6 ^ 6 = 46656 = 1 (mod 31)

--- 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 02:33:55 2022
"henh...@gmail.com" <henhanna@gmail.com> writes:
------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days)

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?

Utterly trivial question, just shove it into any maths program.

A better question would be "Is 211185^54364+54364^211185 prime?"

------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days)

No. That's not how Usenet works.

And you're going over old old old old ground.

These are often called "Leyland Primes" after Paul Leyland of Squeamish Ossifrage fame, or "xyyx" primes for the obvious reason. Thousands of
them are known.

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)
• From Richard Heathfield@21:1/5 to Jonathan Dushoff on Mon Aug 29 04:59:23 2022
On 29/08/2022 4:21 am, Jonathan Dushoff wrote:
I know both "bc" and "factor" as standard linux command-line utilities. I just typed "factor <posted large number>" at the command line.

factor has its limits. For example:

Is 170142368926570945154563387946432179597 prime?

(Rot-13'd spoiler in sig.)

Of course, it's possible that your factor version /can/ handle
the above, but mine can't.

--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
(Vg frrzf abg.)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Jonathan Dushoff@21:1/5 to henh...@gmail.com on Sun Aug 28 20:21:55 2022
I know both "bc" and "factor" as standard linux command-line utilities. I just typed "factor <posted large number>" at the command line.

On Tuesday, August 23, 2022 at 2:17:34 PM UTC-4, henh...@gmail.com wrote:
On Monday, August 22, 2022 at 9:17:59 AM UTC-7, Jonathan Dushoff wrote:

If you have bc, you probably have factor, which will tell you:

2619995643649960903610551432833049: 31 59 8543 59316821 2826822827838696527
i couldn't find Factor( ) or Factorize( ) in
http://sciencesoft.at/bc
http://sciencesoft.at/bc/?lang=en
( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
( 30 ^ x + x ^ 30 ) ---- this a prime only when X=1
On Sunday, August 21, 2022 at 5:35:17 PM UTC-4, Edward Murphy wrote:

On 8/19/2022 10:38 PM, henh...@gmail.com wrote:

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
[spoiler space]

Firing up 'bc' on my home server, I found:

30 ^ 13 = 15943230000000000000
13 ^ 30 = 2619995643649944960380551432833049
Trial and error shows that this is composite, equaling
31 * 84515988504837448503566175252679
= 31 * 59 * 1432474381437922855992647038181
(no other factors > 1 and < 100)

There's also
https://www.calculator.net/factor-calculator.html
but it only works on numbers up to 10^13, whereas we've only reduced things to around 10^30 so far.

If you don't have a calculator handy that can handle numbers this large, then you can also work it out by choosing a potential prime factor, then reducing intermediate computations modulo that prime, e.g.
30 ^ 1 = 30 (mod 31)
30 ^ 2 = 30 * 30 = 900 = 1 (mod 31)
30 ^ 3 = 1 * 30 = 30 (mod 31)
...
30 ^ 13 = 30 (mod 31)
and
13 ^ 1 = 13 (mod 31)
13 ^ 2 = 13 * 13 = 169 = 14 (mod 31)
13 ^ 3 = 14 * 13 = 182 = 27 (mod 31)
13 ^ 4 = 27 * 13 = 351 = 10 (mod 31)
13 ^ 5 = 10 * 13 = 130 = 6 (mod 31)
13 ^ 30 = (13 ^ 5) ^ 6 = 6 ^ 6 = 46656 = 1 (mod 31)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From henhanna@gmail.com@21:1/5 to Phil Carmody on Mon Aug 29 20:22:11 2022
On Saturday, August 27, 2022 at 4:50:07 PM UTC-7, Phil Carmody wrote:
"henh...@gmail.com" <henh...@gmail.com> writes:
------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days)

( 30 ^ 13 + 13 ^ 30 ) ---- Is this a prime number ?
Utterly trivial question, just shove it into any maths program.

thanks for hte comment.... i looked up the etym. of [utterly]

Fermat's Little Theorem. <--- i can't remember if i knew this in college.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)