For the past eight weeks I have worked on the Pomerance-Selfridge-Wagstaff pseudoprime problem. We are charged by PSW to find and factor or prove the non-existence of a base-2 (Fermat) pseudoprime, congruent to 2 or 3 modulo 5, which is also a Fibonacci pseudoprime. That is, our n divides the n+1st Fibonacci number. I have had a few ideas and investigated them. I spent a considerable portion of this project in looking for the desired type of number. I spent time investigating properties of numbers that shared some characteristics. I looked at factorizations, base-2 representations, and Zeckendorff representations of base-2 pseudoprimes, fibonacci pseudoprimes, and these pseudoprimes that are 2 or 3 modulo 5. Another idea has been to search naively through sparse sets of numbers with an above average likelihood of having all the desired characteristics. I read briefly that if a prime p has composite Fibonacci number F_p, then F_p is a Fibonacci pseudoprime. This led me to consider Fibonacci numbers that were themselves Fibonacci pseudoprimes. I found such numbers, however not of the desired kind, but it got me sidetracked into finding 'chains' of primes in the Fibonacci sequence. That is, I looked for primes p with f(p), f(f(p))...all primes, for some length of iteration; it would be interesting to see if we could connect this somehow to Mills' theorem. When I turned my attention to proving the non-existence of a number with the required characteristics I set out to find contradictory necessary conditions. Something got me onto looking at the Fibonacci sequence modulo different numbers. Marc Renault's master's thesis has some great information here. This led me to prove, conditionally, that our sought after number must be squarefree. This isn't so great, 1194649 looks to be the least non-squarefree base-2 pseudoprime, but it's a start. Renault defines a(m) to be the place of the first zero residue in the Fibonacci sequence modulo m. He then reports that a(m)=lcm(a(p_i^e_i)), where the p_i are the prime factors of m. This leads one to calculation of a(p^e), and thus to calculation of a(p). The theorem here is: If t is the largest integer such that a(p^t)=a(p), then a(p^e)=a(p)*p^(e-t). The conjecture is that t=1 for all primes p. On this conjecture it is easy to prove that our number, in fact all Fibonacci pseudoprimes, must be squarefree: the only missing piece of information is that a(m) | n iff m | f(n). More may be concluded from Renault's thesis; this may be worth pursuing. Currently, the roadblock in this area is the calculation of a(p); there is no known formula. Further investigation of a(p) may lead to a Korselt-like condition. Along the lines of searching through sparse sets, I conjectured that Mersenne composites are all base-2 pseudoprimes. I proved this, then later found that this was known to Euler and Fermat. I conducted a search for Fibonacci pseudoprimes among the Mersenne numbers and found none. These searches have as an upper limit numbers prohibitively large for factoring purposes. Searching through Mersenne numbers got me thinking about calculating the order of 2 in Z/mZ for various m. This is trivial for Mersenne numbers: the order of 2 in Z/(2^n-1)Z is obviously n. Other results in this area may give us conditions useful in the context of Fibonacci pseudoprimes. All this got me interested in a few tangential questions: *Are Mersenne numbers all squarefree? *Are there any Mersenne numbers that are also base-3 pseudoprimes? *Are there any Mersenne numbers that are also Carmichael numbers? *What does the order of 2 in Z/nZ look like statistically for composite n? There are many other avenues for pursuit other than what I've already mentioned: *Diagonalizing the "Fibonacci matrix" over Z/mZ *Implementing quicker, yet storage intensive, search algorithms. One such route utilizes the formula: F_(m+10)=L_10 * F_m - F_(m-10), where L_10 is the tenth Lucas number. *Implementing all the searches at much higher search range placements. That is, if pseudoprimes are more common among large numbers, then searching from 10^400 to 10^400+10^5 may be more fruitful than searching from 1 to 10^5. *Implementing the many helpful suggestions: modifying the question to look at different sequences, bases for the Fermat pseudoprime, different residue classes modulo different integers. *Investigating the possible "Euclid-like" generation of solutions. If we find an iterative procedure that will terminate with a solution, then by manipulating some initial values we may be able to generate a number matching all the criteria. In addition, some kind of loose generational structure might appear with the Fibonacci pseudoprimes as there is with the Carmichaels in the form of factor sharing. Basically, we expect some condition very much like the Korselt condition. *I'd like to compile some statistics on what has already been found and what searches I did that came up empty. *I'd like to see if what is known about polynomials and base-b (skew)representations can be brought to bear on problems relating to fibonacci pseudoprimes. *Maybe playing around with some fibonacci identities will be of some use. *In "Prime Numbers" the authors give a proof of the infinitude of Fermat pseudoprimes for each base. For base-2, all the generated pseudoprimes are congruent to 1 ( mod 5); if this can be tweaked to give numbers of the correct residue, it might be of some help. * A list of Carmichael numbers up through 10^15 has been compiled, and this should be checked if this hasn't already been done. * A proper search of the literature would most likely turn up much more than I've found already. * Investigating whether we can make sense of formulas for fibonacci numbers involving nearest- integer expressions in a modular setting. * Investigating any links of Renault's k or a functions to other arithmetic functions. Some of my time has been spent learning how to work with computers better. I have learned some points about efficient coding, different operating systems, and machine architecture. Some of this had direct bearing on the PSW problem, some did not. I'd love to learn this subject matter to investigate efficiently the few ideas that I did have. On this note, all of the programming for this project has been under PARI-GP, a fast number theory software package. I have also spent some time on another question posed in Crandall and Pomerance's "Prime Numbers": are there consective primes r,s,t with r > 61 such that r | st+1. After some thought and conversation with Dr. Lacey, this was tied to a conjecture concerning gaps between primes, which is well out of my reach. After all this, I look forward to returning to this problem later and continuing to learn about Fibonacci numbers, primality testing, and computational issues. I wish I could have spent longer on this and could have had more access to journals, especially the Fibonacci Quarterly. It was instructive and encouraging, though, to be mentored throughout the process.