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nt.quantity principle – Can there breathe a generalization for n within the equation $2^x=a−n$? retort

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nt.quantity principle – Can there breathe a generalization for n within the equation $2^x=a−n$?

This vitality emerge a imprecise query, however let me provide you with its background earlier than happening.
I maintain obtained an equation the place a=2mn+m+n, the place all a,n and m are unaffected numbers and neither ‘m’ or ‘n’ or each of them can breathe equal to 0 (0 will not be a unaffected quantity, however a all quantity).

Not all values of a i.e. the unaffected numbers respond 2mn+m+n and these are the numbers that I necessity to seek out (i.e. the values for ‘a’ which do not respond $2mn+m+m$).

For this, I graze that 2mn+m+n can breathe written as $(2mn+m) + n = (2n+1)m + n$.
All the numbers can breathe represented as (2n+1)m aside from $2^x$.
This is as a result of, in $(2n+1)m$, $2n+1$ is an weird quantity, i.e. all of the weird numbers can written as such.
Similarly, solely these airplane numbers whose prime factorization consists purely of airplane numbers i.e. 2, CANNOT breathe written on this design i.e. $2^x$ numbers.

We maintain some values for ‘a’ which aren’t equal to 2mn+m+n and we’re seeking to discover these.
Similarly, we graze that (2n+1)m too CANNOT breathe written as $2^x$.

Thus, $a ≠ 2mn+m+n$, $2^x ≠ (2n+1)m$ i.e. $2mn+m$.

Therefore, $2^x+n≠ 2mn+m+n$ (the inequality is maintained right here),
$2^x+n ≠ a$

Now, a will not be lonely however, itself is a sever of the equation $2a+1$, the place ‘a’ CAN breathe represented as
2mn+m+n, i.e. ‘2a+1’ is an weird amalgam quantity, as in $2a+1$, a can breathe written as 2mn+m+n, how??
$(2n+1)(2m+1) = 4mn+2m+2n+1 = 2(2mn+m+n)+1$, whereas the values for a which do not respond $2mn+m+n$, $2a+1$ is a
prime quantity.

Therefore, $a ≠ 2^x+n$. Similarly, $2a+1 ≠ 2(2^x+n)+1$ (once more, inequality is maintained).

$2^{x+1} + 2n+1≠2a+1$.
Thus, $2^{x+1} ≠ 2a-2n$,

$2^{x+1} ≠ 2(a-n)$,

$2^x≠a-n$, the place ‘a’ can breathe represented as 2mn+m+n.

i.e. $2^x≠2mn+m+n-n$,

giving us,
$2^x≠2mn+m$,
$2^x≠(2n+1)m$

And, the place ‘a’ CANNOT breathe represented as $2mn+m+n$, (i.e. the place $2a+1$ is prime)

$2^x=a-n$

$2^x$ is given, and there’s a pre requisite situation for $a$.
So, my query –
Is there any path to generalize the values for $n$??

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