Wikipedia:Reference desk/Archives/Mathematics/2022 June 13

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June 13

a^x+b^x=c^x

This video talks about solving the equation 16^x+20^x=25^x. The solution seems to require a special relationship between the bases, in this case that 16⋅25=20². I was wondering if how many other cases exist where x can be given in closed form. The solutions to 3^x+4^x=5^x and 3^x+4^x=7^x are obvious, but 3^x+4^x=6^x apparently not. Do such equations in general have to be solved numerically? Are there qualitatively different cases where a closed form solution is possible? --RDBury (talk) 04:13, 13 June 2022 (UTC)

• The change of variables taken in the video does require a special relationship, but it can be generalised, in such a way that one can prove the general solution cannot be given by radicals (via the Abel–Ruffini theorem). Now, "solvable by radicals" is weaker than "there is a closed form", but still.

Assuming $a\leq b<c$, let $k,y,\alpha ,M,N$ be such that $a=ky^{\alpha },b=ky^{\alpha +M},c=ky^{\alpha +N}$. Such numbers can be found by picking an arbitrary k such that $0<k\neq a$, then defining $y=(a/k)^{1/a}$, then $M=ln(b/a)/ln(y)$ and $N=ln(c/a)/ln(y)$. (The video case is $\alpha =0,M=1,N=2$.) Notice also that N>M.

With that transformation, $a^{x}+b^{x}=c^{x}$ becomes $t^{N}-t^{M}-1=0$ for $t=y^{x}$. That function has only one positive zero (because N>M, the derivative is negative close to zero, and becomes positive at some large t).

Now, according to Quintic_function#Finding_roots_of_a_quintic_equation, $t^{5}-t-1$ is not solvable by radicals. Therefore, if we find $a,b,c$ such that we can apply the previous transform and get M=1, N=5, that case is not solvable by radicals either. Thus, taking k=1, y=2, α=1, I claim that $2^{x}+4^{x}=64^{x}$ cannot be solved by radicals.

I believe that transformation is a useful starting point to solve some cases. Essentially, fiddle with k to make M and N nice values. "Nice values" are those such that there are m, n, β such that $M=m^{\beta },N=n^{\beta }$ with m and n integers between 0 and 4 (because then, $t^{N}-t^{M}-1=0$ becomes $(t^{\beta })^{n}-(t^{\beta })^{m}-1=0$ which is always solvable). I am not sure whether "nice value" pairs are rare or not. Tigraan^{Click here for my talk page ("private" contact)} 09:13, 13 June 2022 (UTC)

• (edit conflict) The solution in the video hinges on the fact that we can give explicit solutions of the equation z² + z = 1. The equation z³ + z = 1 can also be solved algebraically. Let ζ = 0.6823278... denote its real solution – may I be forgiven forgoing writing down the messy algebraic expression. Using this, we can solve the equation 8^x + 18^x = 27^x in closed form by solving ζ = (2⁄3)^x for x.  --Lambiam 09:27, 13 June 2022 (UTC)

Likewise, 4^x + 16^x = 2^x is solved by x = log₂ ζ.  --Lambiam 11:38, 13 June 2022 (UTC)

And, of course, we can somewhat trivially use Pythagorean triples to construct equations like 27^x + 64^x = 125^x that can be solved exactly.  --Lambiam 11:45, 13 June 2022 (UTC)

• Although not explicitly stated, I take it that we are only interested in equations of the form a^x + b^x = c^x in which a, b and c are distinct positive integers that do not share a common divisor and the variable to be solved for, x, is real. The solvable cases found thus far are of two radically different types: those in which x is rational, and those in which it is a surd. For the first we can distinguish two classes:

• (R1) a completely trivial kind represented parametrically by (a, b, c) = (uⁿ, vⁿ, (u+v)ⁿ), solved by x = 1⁄n.
• (R2) an only slightly less trivial class represented parametrically by (a, b, c) = (uⁿ, vⁿ, wⁿ), in which (u, v, w) is a Pythagorean triple. This is solved by x = 2⁄n.

The second type is related to algebraic roots of polynomials of the form z^p + z^q − 1. Without loss of generality we can require that p and q are coprime, and that p > q. If ζ is a real-valued positive root that can be expressed in closed form, we have:

• (S) (a, b, c) = (u^p, u^qv^p−q, v^p), solved by x = log_u⁄v ζ.

For p ≤ 4, each polynomial z^p + z^q − 1 has at least one closed-form real-valued root ζ. I conjectured p ≤ 4 to be also a necessary condition for the existence of a closed-form real-valued root, but it is not:

z⁵ + z − 1 = (z² − z + 1)(z³ + z² − 1).

As far as I can see, this is the only counterexample, but I have no proof of that.  --Lambiam 22:13, 13 June 2022 (UTC)

Thanks, that's about what I was expecting. I wasn't very specific in the wording of the question since I didn't want to rule out anything that I couldn't think of. Solutions involving roots of quintic (or higher) equations seem qualitatively the same as the original problem, though not technically in closed form. Anther special case that easy to solve is when a=b, but it falls in to the type (S) category as well. --RDBury (talk) 02:45, 14 June 2022 (UTC)

Yes, a = b rolls out when the requirement p > q is relaxed to p ≥ q.  --Lambiam 04:30, 14 June 2022 (UTC)

It's a bit cheaty, but using the substitution $z=\alpha ^{n}$ for $n>1$ yields polynomials $\alpha ^{5n}+\alpha ^{n}-1=0$ with real-valued closed-form solutions of the form $\beta ^{\frac {1}{n}}$ where $\beta$ solves $z^{3}+z^{2}-1=0$. Whether all polynomials must be of this form, however, I'm not sure; I'll keep looking. GalacticShoe (talk) 03:33, 14 June 2022 (UTC)

This does not result in new solvable equations, which is why I wrote that we can require without loss of generality that p and q are coprime.  --Lambiam 04:30, 14 June 2022 (UTC)

Oops, missed that part, sorry! GalacticShoe (talk) 16:46, 14 June 2022 (UTC)

As a semi-related conjecture, it seems that $z^{p}+z^{q}-1$ for $p>4$ and $q<p$ is always either irreducible or divisible by $z^{2n}-z^{n}+1$ for some $n$; as to whether this is true or why it might be true, I have no idea. GalacticShoe (talk) 03:42, 14 June 2022 (UTC)

Additionally, if p ≥ 6, the other factor seems to always be irreducible.  --Lambiam 04:48, 14 June 2022 (UTC)

When p and q are coprime, the cases for which z^p + z^q − 1 is divisible by z² − z + 1 appear to be precisely those in which {p, q}  can be written as {6m + 1, 6n + 5}.  --Lambiam 04:56, 14 June 2022 (UTC)

The last part is easy by the root-factor theorem, after considering which pairs of sixth roots of unity add to 1. JBL (talk) 00:54, 16 June 2022 (UTC)