Eliminate the constraint by writing , with . The logarithms cancel, so the objective becomes . Its derivatives are and . Thus the unique stationary point is the unique global minimum, at . Restoring gives
As a direct global check, the arithmetic-geometric mean inequality gives , with equality exactly at the same point.
Put . For , convexity gives , hence
By the first-order form of Taylor's theorem, the right side tends to as . Therefore
This is the supporting inequality for a differentiable convex function. The stated partial derivatives are enough: coordinate derivatives give directional Taylor expansion for convex functions, so no unmentioned continuity assumption on the second partial derivatives is needed. The result in fact needs less smoothness than the question assumes.
The allowed primes are exactly . Define
The fundamental theorem of arithmetic makes this definition unambiguous and proves injectivity. For every , the positive integer belongs to and maps to that tuple, proving surjectivity. The inverse function is explicitly . In particular, maps to . This is the prime-exponent parametrization of smooth numbers.
Fix and use two-base mathematical induction on . At , the proposed right side is ; at , it is . Suppose the identity holds at and . The defining recurrence relation for the Fibonacci numbers gives
This is the assertion at , completing the proof of the Fibonacci addition formula. The argument includes , since is defined.
At , the product is , as required. Suppose the formula holds at . Appending the next factor gives
which is exactly the formula at . Thus mathematical induction gives for every . The factorization also explains the cancellation in this telescoping product.
Use strong induction with three consecutive base cases:
For , assume all values from through have the required representation. Since , write with nonnegative integers. Then is another such representation. Strong induction proves the claim for every . This is a consecutive-base induction for additive semigroups: the three base cases handle all residues before repeated addition of .

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact