# Weak Lower Bound in Apostol's “Number Theory”?

In Apostol's "Introduction to Analytic Number Theory" on page 7 he introduces Fermat numbers of the form $F_n = 2^{2^{n}} + 1$ where $n$ is a non-negative integer. He then states that

The greatest known Fermat composite, $F_{1945}$, has more than $10^{582}$ digits, a number larger than the number of letters in the Los Angeles and New York telephone directories combined.

This is a true statement, but unless I'm misinterpreting the phone book estimate, it falls quite a bit short of capturing the enormity of $2^{2^{1945}}$, which is more than the number of atoms in multiple universes.

The way I am interpreting the phone book estimate is that one counts the number of letters and numbers in the phone book and multiplies them by the population of NY and LA.

Is there a different way to interpret this statement so that it is closer to the value of $F_{1945}$?

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It may be possible that when the book was originally written, the greatest known Fermat composite was smaller, and in later editions/printings the number was changed but not the accompanying comment. (I don't know the publication history of this book.) – Michael Lugo Nov 1 '11 at 16:00
It does say that the book evolved over 25 years as a course at Caltech. The text was published in 1976 so $1976 - 25 = 1951$. Maybe in 1951 the largest Fermat composite wasn't too big. – Unreasonable Sin Nov 1 '11 at 16:24

I think Apostol was just saying that the exponent $2^{1945}$ has more than $10^{582}$ digits.

If we write $2^{1945} = 10^x$ and solve for $x$, we get

$$x = \frac{1945 \log(2)}{\log (10)} \approx 585.5.$$

It should also be noted that this is no longer the largest known composite Fermat number. It has been shown that

• Update: I was incorrect in my reporting of world records: It seems the largest known composite Fermat number is much larger than I previously stated.

$$F_{2141872} = 2^{2^{2141872}} + 1$$

is composite.

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As you point out, the "a number" in Apostol clearly refers to the number of digits, roughly $10^{582}$. Still somewhat off. But some known upper estimates in Number Theory are probably roughly as far away from the truth. – André Nicolas Nov 1 '11 at 16:18

Let' denote:

$a=$number of letters in the Los Angeles telephone directory

$b=$number of letters in the New York telephone directory

So we may interpret statement from the book as:

$F_{1945}>a^{b}$ and $F_{1945}>b^{a}$

Of course one may interpret word "combined" from the statement in some other way since it isn't specified which mathematical operations word "combined" defines.

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