Frank Morgan

June 1, 2000

Old Challenge. Consider the sequence of integers 10ⁿ + 1:

11, 101, 1001, . . .

Are there infinitely many primes? Infinitely many composite numbers?

Answer. High school student Andrew Sullivan was the first to submit that whenever n is odd, 10ⁿ + 1 is divisible by 11. Indeed, modulo 11, 10ⁿ + 1 = (-1)ⁿ + 1 = 0 if n is odd. So there are infinitely many composite numbers.

It seems to be an open question whether there are infinitely many primes, but Luke Gustafson argues against it. The same argument as above shows that whenever n has an odd factor (as 56 = 8 x 7 has the odd factor 7), then 10ⁿ + 1 can be factored. For example, modulo 10⁸ + 1, 10⁵⁶ + 1 = (-1)⁷ + 1 = 0. Therefore for 10ⁿ + 1 to be prime, n must be a power of 2: 10ⁿ + 1 = .

Gustafson continues: “On the other hand,  does not factor algebraically, so it is quite possible that there are infinitely many primes of this form. As one would expect, mathematicians have spent a good deal of time investigating this form. The only known prime values for the expression are  = 10¹ + 1 = 11 and  = 10² + 1 = 101, and there are no more prime values up to and including . It isn’t very likely that there are infinitely many primes; after all,  doubles in digits with each increase in k, thereby halving the chance of primality (by the Prime Number Theorem).”

Also open is the similar question of whether there are infinitely many “Fermat” primes , related to whether a regular N-gon can be constructed with ruler and compass.

Spain. I am currently touring Spain, giving a short course on geometric measure theory and popular talks in Spanish on “The Geometry of Soap Bubbles 2000,” including news on our recent proof of the Double Bubble Conjecture (see Math Chat of March 18). The leading newspaper, El Pais, wrote (as best as I can translate it): “I want to know, for example, how much Morgan’s visit is costing us. . . Unless this gentleman can explain to me how the soap bubbles of 2000 are markedly different from those of 1999, I do not understand why an academic institution would spend money for something like this.” Fortunately there were some good responses. I have tried to explain that geometry provides one way to understand the world and the universe, and that the way to understand the complicated geometry of the universe is to start with the simpler geometry of soap bubbles.

In general, mathematics is flourishing, thanks in part to the efforts of the Spanish Mathematical Society under President Antonio Naveira (Valencia) and Secretary Salvador Segura Gomis (Alicante and Murcia). Gomis contributes the following new challenge:

New Challenge. What is the shortest line segment fencing off prescribed area 0 The next Math Chat will appear on the fourth instead of the third Thursday of June.

Send answers, comments, and new questions by email to [email protected], to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan’s homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.