Q: Why are Buffalo Bills unlike Dollar Bills? A: Dollar Bills are good for 4 quarters

Josh Allen is not appearing in today’s Super Bowl. He led the Buffalo Bills to not just one but two go-ahead touchdowns in the final 2:00 of the game at Kansas City three weeks ago, but the Bills lost both leads and KC won in overtime.

Today we note an inequality that treats the real numbers like a football gridiron.

The two-minute drill is the hallmark of quarterback heroism. KC’s Patrick Mahomes, however, demonstrated the 13-second drill in two plays plus a tying field goal. That is, the Bills were good for 3 quarters plus 14:47. But Mahomes is not in the Super Bowl either. Two weeks ago, he lost the magic touch in both the closing minute of the fourth quarter and the beginning of overtime as KC squandered a big lead and lost to the Cincinnati Bengals. Whose quarterback, Joe Burrow, is playing in today’s Super Bowl, opposite Matthew Stafford of the Los Angeles Rams.

The American football field is divided into 100 yards, but rarely subdivided beyond that. The announcers may refer to “a long two yards” or “a short 3,” but never 2.5 yards. It is not just the announcers. Official statistics are kept in units of whole yards, more often rounded up than down. A fourth-down quarterback plunge with 3 inches to go still counts as a 1-yard gain. Perhaps it is essential to the yard that it not be divided into dyadic or decimal units, the way the meter is. As the US celebrates an event still enjoyed by “one nation indivisible,” we note the indivisible.

Inequalities

Godfrey Hardy, John Littlewood, George Polya are famous for many things separately and together. All three lent their names to the timeless book Inequalities. It codifies the theory of real inequalities.

An Inequality

We recently had reason to look at the following inequality:

If , then

We noted that this fails in general: Let and . Then and

This is not even close. But wait—the following lemma is true:

Lemma 1 If , then

provided are integers.

The proof is quite simple. We note that

But this shows that is a non-zero divisor of . But then

This proves the inequality.

Open Problems

Is this interesting at all? We have an application of the above lemma, which we will discuss in the future. Are there other good examples of inequalities that are general and natural and only hold over the integers? Or is seeking them like trying to “outsmart math itself”?

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