Average liar count for degree-2 Frobenius pseudoprimes

In this paper we obtain lower and upper bounds on the average number of liars for the Quadratic Frobenius Pseudoprime Test of Grantham, generalizing arguments of Erd\H{o}s and Pomerance, and Monier. These bounds are provided for both Jacobi symbol plus and minus cases, providing evidence for the existence of several challenge pseudoprimes.Comment: 19 pages, published in Mathematics of Computation, revised version fixes typos and made a minor correction to the proof of Lemma 18 (result remains unchanged

Glimpse on the relationship between Feynman integral and integrable system

We briefly review recent attempts to relate the concept of Feynman integral and integrable systems. This constitutes an endeavour on our part in making the Feynman path integral into a mathematically meaningful entity