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tech / sci.math / Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

SubjectAuthor
* Re: Two whole-numbers A and B that satisfies A^B = B^A, and aChris M. Thomasson
+* Re: Two whole-numbers A and B that satisfies A^B = B^A, and aChris M. Thomasson
|`* Re: Two whole-numbers A and B that satisfies A^B = B^A, and aChris M. Thomasson
| `- Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if FromTheRafters
`* Re: Two whole-numbers A and B that satisfies A^B = B^A, and aVolney
 `* Re: Two whole-numbers A and B that satisfies A^B = B^A, and aChris M. Thomasson
  `- Re: Two whole-numbers A and B that satisfies A^B = B^A, and aChris M. Thomasson

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Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Chris M. Thomasson - Sat, 4 Nov 2023 07:02 UTC

On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
> Alright, I am closing in on, or converging in math-speak to a solution.
>
> I like to use 3 as for 3rd dimension, and like to use one of the "highly geometrical constants" like 3.14... or square root of 10 as 3.162.... or 2.71... the equiangular log spiral or even 1.618... the Golden Mean ratio.
>
> So trying 3 with a version of 2.71... and using the fraction version of Golden Mean as 2.618....
>
> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>
> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead of 0.20
>
> So somewhere nearby 3 and 2.618 is a convergence point.
>
> AP

A = 1
B = 1

1^1 = 1^1

?

Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Chris M. Thomasson - Sat, 4 Nov 2023 07:17 UTC

On 11/4/2023 12:11 AM, Archimedes Plutonium wrote:
> On Saturday, November 4, 2023 at 2:03:03 AM UTC-5, Chris M. Thomasson wrote:
>> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>>> Alright, I am closing in on, or converging in math-speak to a solution.
>>>
>>> I like to use 3 as for 3rd dimension, and like to use one of the "highly geometrical constants" like 3.14... or square root of 10 as 3.162.... or 2.71... the equiangular log spiral or even 1.618... the Golden Mean ratio.
>>>
>>> So trying 3 with a version of 2.71... and using the fraction version of Golden Mean as 2.618....
>>>
>>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>>
>>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead of 0.20
>>>
>>> So somewhere nearby 3 and 2.618 is a convergence point.
>>>
>>> AP
>> A = 1
>> B = 1
>>
>> 1^1 = 1^1
>>
>> ?
>
> Dufus dolt chris A is different from B. But a jackarse like you needs everything spelled out. That is why you failed Kindergarten

You said two whole numbers. Where did you say they must be different?

Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Chris M. Thomasson - Sat, 4 Nov 2023 07:38 UTC

On 11/4/2023 12:27 AM, Archimedes Plutonium wrote:
> On Saturday, November 4, 2023 at 2:17:34 AM UTC-5, Chris M. Thomasson wrote:
>> On 11/4/2023 12:11 AM, Archimedes Plutonium wrote:
>>> On Saturday, November 4, 2023 at 2:03:03 AM UTC-5, Chris M. Thomasson wrote:
>>>> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>>>>> Alright, I am closing in on, or converging in math-speak to a solution.
>>>>>
>>>>> I like to use 3 as for 3rd dimension, and like to use one of the "highly geometrical constants" like 3.14... or square root of 10 as 3.162.... or 2.71... the equiangular log spiral or even 1.618... the Golden Mean ratio.
>>>>>
>>>>> So trying 3 with a version of 2.71... and using the fraction version of Golden Mean as 2.618....
>>>>>
>>>>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>>>>
>>>>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead of 0.20
>>>>>
>>>>> So somewhere nearby 3 and 2.618 is a convergence point.
>>>>>
>>>>> AP
>>>> A = 1
>>>> B = 1
>>>>
>>>> 1^1 = 1^1
>>>>
>>>> ?
>>>
>>> Dufus dolt chris A is different from B. But a jackarse like you needs everything spelled out. That is why you failed Kindergarten
>> You said two whole numbers. Where did you say they must be different?
>
> Just like your idiot criminal accomplice Dan Christensen with his "unique two points". Given any two points. If I say given any two points in geometry, only a lunatic like you Chris and Dan would not understand that it cannot be one point.
>
> When someone with at least 1 marble of Logic intelligence says TWO Whole Numbers, understands that 1 is just one whole number.
>
> No wonder you work for the UK government, you are too dumb to navigate the world without agents hand walking you around London. Do they breast feed you too Chris.
>
> AP

Two whole numbers.

A = 1
B = 1

Well, A and B are two whole numbers.

Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist
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 by: FromTheRafters - Sat, 4 Nov 2023 09:51 UTC

Chris M. Thomasson pretended :
> On 11/4/2023 12:27 AM, Archimedes Plutonium wrote:
>> On Saturday, November 4, 2023 at 2:17:34 AM UTC-5, Chris M. Thomasson
>> wrote:
>>> On 11/4/2023 12:11 AM, Archimedes Plutonium wrote:
>>>> On Saturday, November 4, 2023 at 2:03:03 AM UTC-5, Chris M. Thomasson
>>>> wrote:
>>>>> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>>>>>> Alright, I am closing in on, or converging in math-speak to a solution.
>>>>>>
>>>>>> I like to use 3 as for 3rd dimension, and like to use one of the
>>>>>> "highly geometrical constants" like 3.14... or square root of 10 as
>>>>>> 3.162.... or 2.71... the equiangular log spiral or even 1.618... the
>>>>>> Golden Mean ratio.
>>>>>>
>>>>>> So trying 3 with a version of 2.71... and using the fraction version of
>>>>>> Golden Mean as 2.618....
>>>>>>
>>>>>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>>>>>
>>>>>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead of
>>>>>> 0.20
>>>>>>
>>>>>> So somewhere nearby 3 and 2.618 is a convergence point.
>>>>>>
>>>>>> AP
>>>>> A = 1
>>>>> B = 1
>>>>>
>>>>> 1^1 = 1^1
>>>>>
>>>>> ?
>>>>
>>>> Dufus dolt chris A is different from B. But a jackarse like you needs
>>>> everything spelled out. That is why you failed Kindergarten
>>> You said two whole numbers. Where did you say they must be different?
>>
>> Just like your idiot criminal accomplice Dan Christensen with his "unique
>> two points". Given any two points. If I say given any two points in
>> geometry, only a lunatic like you Chris and Dan would not understand that
>> it cannot be one point.
>>
>> When someone with at least 1 marble of Logic intelligence says TWO Whole
>> Numbers, understands that 1 is just one whole number.
>>
>> No wonder you work for the UK government, you are too dumb to navigate the
>> world without agents hand walking you around London. Do they breast feed
>> you too Chris.
>>
>> AP
>
> Two whole numbers.
>
> A = 1
> B = 1
>
> Well, A and B are two whole numbers.

Right. Most mathematicians would avoid the confusion by being explicit.
Something like:

Two whole numbers A and B with A not equal to B.
Two distinct (unique, different, separate, etc...) whole numbers A and
B.

Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: vol...@invalid.invalid (Volney)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Volney - Sat, 4 Nov 2023 20:57 UTC

On 11/4/2023 3:02 AM, Chris M. Thomasson wrote:
> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>> Alright, I am closing in on, or converging in math-speak to a solution.
>>
>> I like to use 3 as for 3rd dimension, and like to use one of the
>> "highly geometrical constants" like 3.14... or square root of 10 as
>> 3.162.... or 2.71... the equiangular log spiral or even 1.618... the
>> Golden Mean ratio.
>>
>> So trying 3 with a version of 2.71... and using the fraction version
>> of Golden Mean as 2.618....
>>
>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>
>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead of
>> 0.20
>>
>> So somewhere nearby 3 and 2.618 is a convergence point.
>>
>> AP
>
> A = 1
> B = 1
>
> 1^1 = 1^1
>
> ?

Yes, all x^x, where defined, are trivial solutions. For x != y, the only
integer solutions are (2,4) and (-2,-4). There is a hyperbola-like curve
of solutions, with values such as (2.5,2.970287...) and
(3,2.47805268...) on it as well as (2,4).

Re: Two whole-numbers A and B that satisfies A^B = B^A, and a possible proof if none exist

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Chris M. Thomasson - Sat, 4 Nov 2023 21:28 UTC

On 11/4/2023 1:57 PM, Volney wrote:
> On 11/4/2023 3:02 AM, Chris M. Thomasson wrote:
>> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>>> Alright, I am closing in on, or converging in math-speak to a solution.
>>>
>>> I like to use 3 as for 3rd dimension, and like to use one of the
>>> "highly geometrical constants" like 3.14... or square root of 10 as
>>> 3.162.... or 2.71... the equiangular log spiral or even 1.618... the
>>> Golden Mean ratio.
>>>
>>> So trying 3 with a version of 2.71... and using the fraction version
>>> of Golden Mean as 2.618....
>>>
>>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>>
>>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead
>>> of 0.20
>>>
>>> So somewhere nearby 3 and 2.618 is a convergence point.
>>>
>>> AP
>>
>> A = 1
>> B = 1
>>
>> 1^1 = 1^1
>>
>> ?
>
> Yes, all x^x, where defined, are trivial solutions. For x != y, the only
> integer solutions are (2,4) and (-2,-4).

Humm... I feel like using 2+4i and -2-4i for one of my multi-julias.
Here is my basic algorithm:

https://paulbourke.net/fractals/multijulia

I wonder what its going to look like. I will keep you posted.

There is a hyperbola-like curve
> of solutions, with values such as (2.5,2.970287...) and
> (3,2.47805268...) on it as well as (2,4).

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From: chris.m....@gmail.com (Chris M. Thomasson)
Newsgroups: sci.math
Subject: Re: Two whole-numbers A and B that satisfies A^B = B^A, and a
possible proof if none exist
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 by: Chris M. Thomasson - Sat, 4 Nov 2023 21:45 UTC

On 11/4/2023 2:28 PM, Chris M. Thomasson wrote:
> On 11/4/2023 1:57 PM, Volney wrote:
>> On 11/4/2023 3:02 AM, Chris M. Thomasson wrote:
>>> On 11/3/2023 11:22 PM, Archimedes Plutonium wrote:
>>>> Alright, I am closing in on, or converging in math-speak to a solution.
>>>>
>>>> I like to use 3 as for 3rd dimension, and like to use one of the
>>>> "highly geometrical constants" like 3.14... or square root of 10 as
>>>> 3.162.... or 2.71... the equiangular log spiral or even 1.618... the
>>>> Golden Mean ratio.
>>>>
>>>> So trying 3 with a version of 2.71... and using the fraction version
>>>> of Golden Mean as 2.618....
>>>>
>>>> 2.618^3 = 17.94 and then 3^2.618 = 17.74 and so I am off by 0.20.
>>>>
>>>> Now if I use 2.99 instead of 3, and 2.618, I am off by 0.18 instead
>>>> of 0.20
>>>>
>>>> So somewhere nearby 3 and 2.618 is a convergence point.
>>>>
>>>> AP
>>>
>>> A = 1
>>> B = 1
>>>
>>> 1^1 = 1^1
>>>
>>> ?
>>
>> Yes, all x^x, where defined, are trivial solutions. For x != y, the
>> only integer solutions are (2,4) and (-2,-4).
>
> Humm... I feel like using 2+4i and -2-4i for one of my multi-julias.
> Here is my basic algorithm:
>
> https://paulbourke.net/fractals/multijulia
>
> I wonder what its going to look like. I will keep you posted.
>
>
> There is a hyperbola-like curve
>> of solutions, with values such as (2.5,2.970287...) and
>> (3,2.47805268...) on it as well as (2,4).
>
>
>
>

Here it is:

https://i.ibb.co/hFN5nQJ/image.png

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