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From: William Elliot on 1 Jul 2010 04:39
On Wed, 30 Jun 2010, Deep wrote:

> Consider the following two equations under the given conditions.
>
> k^2 = (a^k + b^k)/u^2 (1)
>
> k^2 = [c^k +(1/4)d^k]/v^2 (2)
>
> Conditions: All the variables are real and each > 1; a, b are integers
> and coprime,
> b is even; c, d are coprime and d is even; prime k >
> 3.
>
> Conjecture: If (1) is impossible when u is an odd integer then (2) is
> also impossible when v is an odd integer.
>
Is this below a correct simplification of your conjecture?
Is the statement below equivalent to your proposition?

For prime k > 3,
if some coprime a,b in N\1, n in N with
4(2n + 1)^2 k^2 = 4a^k + (2b)^k, (2)

then some coprime a,b in N\1, n in N with
(2n + 1)^2 k^2 = a^k + (2b)^k. (1)

> Any comment upon the validity of the conjecture will be appreciated.
>
I've spend my time simplifying your proposition instead of thinking about
it. Simplify your propositions, then I'll have time to think about them.
Anyway, unless I hear from you that I've correctly simplified your
proposition, to give it further thought could be a waste of time.

From: William Elliot on 2 Jul 2010 00:30
On Thu, 1 Jul 2010, William Elliot wrote:
> On Wed, 30 Jun 2010, Deep wrote:
>
>> Consider the following two equations under the given conditions.
>>
>> k^2 = (a^k + b^k)/u^2 (1)
>>
>> k^2 = [c^k +(1/4)d^k]/v^2 (2)
>>
>> Conditions: All the variables are real and each > 1; a, b are integers
>> and coprime,
>> b is even; c, d are coprime and d is even; prime k >
>> 3.
>>
>> Conjecture: If (1) is impossible when u is an odd integer then (2) is
>> also impossible when v is an odd integer.
>>
> Is this below a correct simplification of your conjecture?
> Is the statement below equivalent to your proposition?
>
> For prime k > 3,
> if some coprime a,b in N\1, n in N with
> 4(2n + 1)^2 k^2 = 4a^k + (2b)^k, (2)
>
> then some coprime a,b in N\1, n in N with
> (2n + 1)^2 k^2 = a^k + (2b)^k. (1)

If prime k > 3, coprime a,b in N\1, n in N, and
4(2n + 1)^2 k^2 = 4a^k + (2b)^k, (2)

then some coprime a,b in N\1, n in N with
(2n + 1)^2 k^2 = a^k + (2b)^k. (1)

--
If prime k > 3, coprime a,b in N\1, n in N, and
(2n + 1)^2 k^2 = a^k + 2^(k-2) b^k, (2)

then some coprime a,b in N\1, n in N with
(2n + 1)^2 k^2 = a^k + 2^k b^k. (1)

>> Any comment upon the validity of the conjecture will be appreciated.
>>
> I've spend my time simplifying your proposition instead of thinking about it.
> Simplify your propositions, then I'll have time to think about them. Anyway,
> unless I hear from you that I've correctly simplified your proposition, to
> give it further thought could be a waste of time.
>
>
From: William Elliot on 3 Jul 2010 03:30
On Thu, 1 Jul 2010, William Elliot wrote:
> On Thu, 1 Jul 2010, William Elliot wrote:
>> On Wed, 30 Jun 2010, Deep wrote:
>>
>>> Consider the following two equations under the given conditions.
>>>
>>> k^2 = (a^k + b^k)/u^2 (1)
>>>
>>> k^2 = [c^k +(1/4)d^k]/v^2 (2)
>>>
>>> Conditions: All the variables are real and each > 1; a, b are integers
>>> and coprime,
>>> b is even; c, d are coprime and d is even; prime k >
>>> 3.
>>>
>>> Conjecture: If (1) is impossible when u is an odd integer then (2) is
>>> also impossible when v is an odd integer.
>>>
>> Is this below a correct simplification of your conjecture?
>> Is the statement below equivalent to your proposition?

> If prime k > 3, coprime a,b in N\1, n in N, and
> 4(2n + 1)^2 k^2 = 4a^k + (2b)^k, (2)
>
> then some coprime a,b in N\1, n in N with
> (2n + 1)^2 k^2 = a^k + (2b)^k. (1)
>
> --
> If prime k > 3, coprime a,b in N\1, n in N, and
> (2n + 1)^2 k^2 = a^k + 2^(k-2) b^k, (2)
>
> then some coprime a,b in N\1, n in N with
> (2n + 1)^2 k^2 = a^k + 2^k b^k. (1)
>
-- weaker conjecture
If prime k > 3, coprime a,b in N\1 and
sqr(a^k + 2^(k-2) b^k) in N, (2)

then some coprime a,b in N\1 with
sqr(a^k + 2^k b^k) in N. (1)

--
>>> Any comment upon the validity of the conjecture will be appreciated.
>>>
>> I've spend my time simplifying your proposition instead of thinking about
>> it. Simplify your propositions, then I'll have time to think about them.
>> Anyway, unless I hear from you that I've correctly simplified your
>> proposition, to give it further thought could be a waste of time.
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