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Zack Wolske's user avatar
Zack Wolske's user avatar
Zack Wolske
  • Member for 13 years, 2 months
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  • Toronto, ON, CAN
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
There is a gap here where my argument assumes that no higher powers of a primitive divides its Fibonacci number. As far as I know, it's not resolved.
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
If you're allowing negative numbers, then $F_{36} - F_{-12} = 3864^2$. This comes from writing $F_{n + 4m} - F_n = F_{2m}L_{n+2m}$ and splitting $F_{2m} = F_mL_m$, then taking $F_m$ to be a square and setting $m= n+2m$. As before, this only works because $F_m$ is a square, which only happens a few times. Also note, allowing negatives is the same as asking "When is $F_{2n + m} + F_{2n}$ a square?", since we only duplicate our positive solutions if $F_{-n} = F_n$.
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
In a similar fashion, you can show that there is a solution (and exactly one) separated by 6 terms: $F_{15} - F_9 = 576 = 24^2$. But this example relies on the Lucas number $L_3$ being a square, so there's no hope that it produces an infinite family.
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Can the difference of two distinct Fibonacci numbers be a square infinitely often?
Using Cohn's characterization of when $F_n = 2x^2$ (it's true only when n = 0, 3, or 6), and simplifying $F_{n+3} - F_n = 2F_{n+1}$, the only examples separated by 3 are $F_5 - F_2 = 4$, and $F_8 - F_5 = 16$. Similarly, to find all pairs separated by 4, simplify $F_{n+2} - F_{n-2} = F_{n+1} + F_{n-1} = L_n$ to get that the Lucas number $L_n$ must be a square, so n = 1 or 3, giving $F_5 - F_1 = 4$ as the only solution. Both characterizations come from J. Cohn's paper "Square Fibonacci Numbers, Etc." Fibonacci Quarterly 2 1964, pp. 109-113.
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Short Course Suggestions For High School Students
Winning Ways is fantastic for combinatorial number theory, because everything is built up through games which anyone can play, and numbers only come in as a notation to keep track of who is ahead in the game. Plus the "numbers" include surreal numbers like stars and arrows. I remember seeing this during a math summer camp when I was 15 and being shocked that something that wasn't a number (like "double up star") could be used to count things, and that those numbers could be combined (added) in a meaningful way.
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A book in topology
Additionally, further courses in algebraic topology can continue using Hatcher. It's nice to get used to his writing style early.
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First correct proof of FLT for exponent 3?
Harold Edwards, in "Fermat's Last Theorem" (Spring-Verlag, 1977) also notes that Euler's original proof via descent fails for the above reasons, and gives a proof that cubes must factor into cubes by using methods known to Euler (arguments Euler used to show that primes congruent to 1 mod 3 can be written as $a^2 + 3b^2$), but also explicitly states that though Euler could have made this argument, he did not.
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covering projection
Have you tried math stackexchange?
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