5. PROJECT 3

Consider the following recursively defined sequence of " $J$ numbers":

\(\left\{\begin{array}{l}

J(0)=1 \\

J(1)=2 \\

J(2)=3 \\

J(n)=J(n-1)+J(n-2)+J(n-3) \text { for } n \geq 3

\end{array}\right.\)

1. Write a set of Mathematica statements to recursively evaluate \(J(n)\) for positive integers \(n\). What are the values of \(J(6), J(36)\), and \(J(100)\) ?
2. By generating appropriate tables using Mathematica, what properties can you conjecture about the \(J\) -numbers. For example: Which \(J\) -numbers are even? Which \(J\) -numbers are divisible by 3 ? Which \(J\) -numbers are perfect squares? Which \(J\) -numbers are perfect cubes? Write your conjectures as clear mathematical statements and then try to provide a proof for each statement to elevate your conclusion to a theorem.
3. Using the Mathematica PrimeQ function (or otherwise), construct a table of \(J\) -number values showing whether each number is prime or not.
   What are the first eight prime \(J\) -numbers and what are their values?
   Define the \(J\) -ratio (for all integers \(n>0\) ) by:
   \(\varphi(n)=\frac{J(n)}{J(n-1)}\) and let \(\rho=\lim _{n \rightarrow \infty} \varphi(n) .\)
4. Obtain an estimate for the numeric value of \(\rho\) correct to 5 decimal places.
5. What are the six most significant digits of \(\rho\) ?
6. Use the \(J\) -number recurrence definition to derive a polynomial expression for the \(J\) -ratio limit \(\rho\). Using the Mathematica Solve command (or otherwise) find all the real zeros of the polynomial you derive, verify that one of these zeros has the approximate value you estimated in (4), and then write down an exact representation of the value of the limit.

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