MHB How can we use hints to prove divisibility of Fibonacci numbers?

  • Thread starter Thread starter evinda
  • Start date Start date
AI Thread Summary
To prove that 30290 divides the Fibonacci number F_m, where m=n^13-n and n>1, it is essential to first establish that a^13 ≡ a (mod 2730). This is shown using Fermat's Little Theorem for the prime factors of 2730, confirming that 2730 divides a^13-a. The next step involves leveraging the relationship that if n divides m, then F_n divides F_m. Participants suggest that additional hints or properties may be necessary to connect the established modular equivalence to the divisibility of F_m by 30290. The discussion emphasizes the need for further exploration of Fibonacci properties in relation to divisibility.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

I want to show that if $m=n^{13}-n$ and $n>1$ then $30290 \mid F_m$. (Hint: Show first that $a^{13} \equiv a \mod{2730}$.)

$F_m$ is the $m$-th Fibonacci number.I have shown the hint as follows:

$2730=2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$.

Using Ferma's little theorem, we deduce that $a^{13}\equiv a \pmod{5}$, $a^{13}\equiv a \pmod{2}$, $a^{13}\equiv a \pmod{3}$, $a^{13}\equiv a \pmod{7}$ and $a^{13}\equiv a \pmod{13}$.Since $2,3,6,7,13$ are all relatively prime, we deduce that $2730 \mid a^{13}-a$.

But how can we use the fact that $a^{13} \equiv a \mod{2730}$ in order to deduce that $30290 \mid F_m$ ? (Thinking)
 
Mathematics news on Phys.org
Hey evinda! (Wave)

I think we need another hint.
Something like $n\mid m \Rightarrow F_n \mid F_m$.
Can we use that? Or something else? (Wondering)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top