## e-day, or why cooking and math don’t mix

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Tomorrow, 2/7, is the day we celebrate e-day, in honor of the number e≈2.7182…. And celebrate we do: there are decorations, and e-related foods (browniEs, e-clairs, pi(e), etc.) and an e-day quiz. So it’s a big deal around here. And sometimes dangerous.

The date was February 7, 2005. The time was 5:00am. I’d gotten up early to cook, and this year decided to try a recipe that involved baking the brownies, then putting white chocolate chips and crushed candy canes on top. I placed the whole pan under the broiler for a few minutes for the chips to soften before spreading them around. And I’m sure all would have gone well if TwoPi hadn’t asked from the dining room, “Hey, are there any nonzero integers a, b, and c such that $a^2+b^3=c^4$?” Naturally I started to work on that question, and forgot all about the brownies.

Until the fire alarm went off.

The candy canes had caught on fire, and the entire pan was in flames (see the evidence in the picture above). We got to try out the fire extinguisher, and managed to get everything under control with only one lost pan of food.

Then TwoPi found a solution to the problem while I cleaned up. It looks like there are an infinite number of answers: if you rearrange the terms you find

$b^3 = c^4 - a^2 = (c^2-a)(c^2+a).$

By setting $(c^2-a)$ equal to 1, the term $(c^2+a)$ simplifies to $(2c^2-1)$ and on a calculator or spreadsheet you can quickly scan through values of c that lead to that term having an integer cube root. For example, $6083^2+23^3=78^4$. Presumably there are other solutions if $(c^2-a)$ is set equal to a different integer. Hmmm….something for the e-day quiz.

### 5 Responses to “e-day, or why cooking and math don’t mix”

1. TwoPi Says:

Indeed, any individual solution will lead to an infinite family of solutions by scaling (multiply both sides of the equation by any perfect 12th power…).

I’ve done some numerical experiments with different values of $c^2 - a$, but have yet to see patterns or generate conjectures.

Poking around Sloane’s On-line Encyclopedia of Integer Sequences for related sequences (values of $c^2 - a$ where solutions exist, or of c, or etc.) came up empty.

Dickson’s History of the Theory of Numbers mentioned one related result involving triangular numbers, but I want to track down the source before commenting further on that. (And Dickson’s History… is huge — I’m not confident that I didn’t miss something else there.)

2. Alon Levy Says:

Actually, there are only finitely many solutions to b^3 = 2c^2 – 1, by Siegel’s theorem on integer points on elliptic curves. In general, any equation in two variables of degree higher than two that satisfies certain smoothness conditions will have finitely many integer solutions.

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