Intereting Posts

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Evaluate limit of $(2\sin x\log \cos x + x^{3})/x^{7}$ as $x \to 0$
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Need help finding the limit of this sequence.
Pollard-Strassen Algorithm
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Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?
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Show that these two numbers have the same number of digits
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Why is this weighted least squares cost function a function of weights?

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is fundamentally different than other problems, similar in the sense of formulation and mentioned in the question I linked to. Naturally, I hope somebody will see something that I don’t, and help deal with this, and share the joy of discovery. The problem is following:

Prove or disprove: If $F_{n}$ is the $n^{th}$ Fibonacci number then $${F_{n}^2} + 43$$

cannot be a prime, except for two cases of $n$.

- Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$
- Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
- How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
- Solution to $x^n=a \pmod p$ where $p$ is a prime
- Direct proof that $n!$ divides $(n+1)(n+2)\cdots(2n)$
- How to prove this curiosity that has to do with cubes of certain numbers?

I checked cases for $n$ up to few thousand, and there was no a prime, except 2 occurrences of small $n$.

Note that smallest divisors of these numbers have a variety of values, and this means that solution that relies on conguences only are unlikely.

Also, in many cases, the numbers ${F_{n}^2} + 43$ are products of only 2 primes, and there are cases where one divisor is small, another large, but also there are cases where both such divisors are approximate in value. This unfortunately indicates that it is very unlikely the proof can rely on some kind of identity involving Fibonacci numbers.

I suspect this problem is either very difficult to prove (that all such numbers are composite), or there is a counter-example for some very large $n$ (lets say $n$ is a million or so).

Warning: *This may be a very difficult problem.*

- Elementary Number Theory Congruence Proof
- Prove that $a+b$ is a perfect square
- Diophantine equation $a^2+b^2=c^2+d^2$
- Show that $(\sqrt{2}-1)^n$ is irrational
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- Inclusion-Exclusion Convergence
- Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$
- Bounds for Waring's Problem
- Find a continuous function $f:[1,\infty)\to\Bbb R $ such that $f(x) >0 $, $\int_1^\infty f(x)\,dx $ converges and $\int_1^\infty f(x)^2\,dx$ diverges

Note that $F_{1863}^2 + 43$ is a prime of $\approx$ $800$ digits.

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