------ 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.

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)

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 ?
>
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> [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

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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)

On 22/08/2022 5:17 pm, Jonathan Dushoff wrote:
>
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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

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)

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)

"henh...@gmail.com" <henhanna@gmail.com> writes:
> ------ pls (pls, Please... PLEASE ! ) wait 3+ days (pref. 10+ days)
> before posting answers or hints.
>
>
>
> ( 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)
> before posting answers or hints.

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/

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)

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.)

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)
> > before posting answers or hints.
> >
> >
> > ( 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.