Defining the Erdős Coefficient

When speaking of this century's most prolific mathematician, Paul Erdős, Alfréd Rényi once said that a "mathematician is a machine for turning coffee into theorems". [1] I closely live by this rule, usually sipping on 2-3 cups per day while doing math and, recently, coding as well.

It's unclear, however, how exactly my coffee intake affects my performance. Some days I have too much and I get a headache. Other days I have too little and thus can't think clearly. And even other days all I need is that sweet first sip in the morning when I arrive at my desk and the lights are dim with only one other student is in the graduate offices with me. Perfect way to start the day.

Therefore, in order to keep track of my productivity, and in honor of Paul Erdős, I wish to define and track my Erdős Coefficient:

Definition: The Erdős Coefficient is the ratio of the number of correct proofs you complete to the number of cups of coffee you drink.

First, a couple of clarifications:

1. By completed proof we mean a correct proof of any kind of problem; whether it's for a course, research, or just for fun. That is, propositions, lemmas, and theorems are considered equal. (Of course, this means you can cheat and declare every line in a proof as a lemma but that's not very fun, now is it?)
2. By "cup" we mean one United States customary cup. This unit of measurement is equivalent to...
• ...8 U.S. customary fluid ounces.
• ...approximately 237mL. [2]
• ...six shot glasses. [3]
• ...about what a standard coffee cup can hold.
• ...a "short" at most coffee stands.

Based on this definition we can make a couple of observations. The first and most obvious is that the Erdős Coefficient is undefined for people who don't drink any coffee. Sorry, but even though I greatly enjoy a cup of tea along with my work I have to exclude such people who care not for coffee's sensual flavors.

Second, if one is able to prove one statement per cup of coffee then they have an Erdős Coefficient of one. This seems like a reasonable "standard" for most graduate students since, at least in my experience in graduate modern algebra, a good portion of exercises take approximately 1-2 hours to figure out whereas a gross estimate on my part suggests that it takes roughly the same amount of time to enjoy a cup of coffee without any ill effects.

Therefore, if one is not making use of their coffee and is thus not proving as many theorems as they should then their Erdős Coefficient approaches zero. Conversely, if one uses their coffee effectively then their Erdős Coefficient can become arbitrarily large. The fact the the value is unbounded above encourages eager graduate students to work until exhaustion; bumping up that value as much as possible. I've found that the key component to increasing, say, a video game's re-playability is to leave the concept of completion as open-ended as possible. So speaking of which, I should wrap up this post as quickly as possible so that I can get back to work.

Finally, used the term "coefficient" since a mathematician's total productivity is not based solely on the number of theorems they prove. Rather, it's a combination of talks given, courses taught, line of code written, and many other factors. So if we let E be the Erdős Coefficient and P be our total productivity then the Erdősian Productivity or Mathematician's Total Productivity, M, is defined

M = EP.

Please post if you have any comments or suggestions about this new number!