Welcome to the 161th Carnival of Mathematics. This is a monthly digest of selected mathematical blogs, hosted each month on a different site. The 1st Carnival has been published in February 2007, so this tradition already continues for more than 11 years.

Following the Carnival’s tradition, first we collect some interesting facts about 161 = 7 x 23. First, it is an generalised octagonal number, and hence 3*161+1 is a perfect square. Next, it is also a greengrocer’s number. Finally, it is the number of isomorphism classes of groups of order 3080.

Which of these facts you would find most interesting? What is your favourite mathematical fact? Evelyn Lamb interviewed two mathematicians about their favourite theorems for her Roots of Unity blog. Holly Krieger chose the Brouwer fixed-point theorem, and Vidit Nanda – the Banach’s fixed-point theorem.

Other blog writers turned to mathematical facts that are not intuitively clear. For example, Elias Wirth used graph theory to describe the phenomenon called the Friendship Paradox: on average, most individual’s friends have more friends than the individual themselves. His blog, called Math Section, is dedicated to applications of mathematic in everyday life, and the friendship paradox is a perfect fit. In particular, the article refers to a study which applies this paradox to better predict outbreaks of contagious diseases.

While the friendship paradox’s explanation is accepted, that’s not yet the case for the proof of the abc conjecture, one of the central modern problems in number theory. You can read about the latest developments in the article by Erica Klarreich called “Titans of Mathematics Clash Over Epic Proof of ABC Conjecture” and in the post by Rachel Crowell called “Musings on a mathematicians duties“, placing the problem in the wider context of research ethics.

Chistina Heimken from the University of Münster interviewed her colleague Raimar Wulkenhaar who, together with Erik Panzer (Oxford) solved an equation from elementary particle physics, previously considered to be unsolvable. The interview reflects on 10 year long attempt to find a solution, and gives an interesting insight into the process of mathematical research.

But mathematical research is inseparable from learning and teaching mathematics. How we combine building theories and solving problems when we teach mathematics?Joshua Bowman looks at this from the philosophical viewpoint in “Dialectics of Mathematics“.

Continuing teaching theme, look also at “An Introduction to Newton’s Method” by Ari Rubinsztejn for geometric explanations and a visualisation of the algorithm.

John D. Cook‘s new post “Pi primes” was inspired by another blog post by Evelyn Lamb in which she had mentioned that 314159 is a prime number. Can we find another prime number formed by the initial digits of π? There must be an OEIS sequence for that! In his another recent post on statistics called “Six sigma events” John discusses the rarity of six-sigma events and concludes that they are much more common than the name implies.

Finally, Q is for QUESTION: especially for this issue of the Carnival, Peter Cameron wrote a blog post called “Q is for quantum?” about several standard uses of the letter q in mathematics.

Thank you to everyone who made contributions to the 161th Carnival! The next 162th Carnival of Mathematics is hosted by the team at the Chalkdust Magazine. Please see the main Carnival website for further details and the form to submit blog posts to the new Carnival.