Euclid and the Infinitude of PrimesAuthor(s): WILLIAM DUNHAMSource: The Mathematics Teacher, Vol. 80, No. 1 (JANUARY 1987), pp. 16-17Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27965215 .

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Euclid and the

Infinitude of Primes By WILLIAM DUNHAM, Hanover College, Hanover, IN 47243

Alas,

poor Yorick! I knew him well.

Many people would identify these poignant lines as having been spoken by Hamlet in the graveyard, just as many people would

recognize Moby Dick as Herman Melville's

masterpiece about a demented sea captain and a great white whale.

That people would make these identifi cations comes in spite of the fact that Hamlet actually uttered these slightly dif ferent lines (Act V, Scene 1) :

Alas, poor Yorick! I knew him, Horatio.

And Melville, in the wry words of Richard Armour (1960, 9), died "blissfully unaware

that, in the years to come, so many people would leave the hyphen out of Moby-Dick.'9

These examples serve as reminders that inaccuracies can creep into the public's consciousness and that that which is popu larly accepted and oft-repeated need not be

entirely genuine. So it is with one of the true masterpieces of mathematics?

Proposition 20 of Book IX of Euclid's Ele ments. In what is commonly known as Euclid's proof of the infinitude of primes, we find a logical gem that the esteemed British mathematician G. H. Hardy (1969, 92) called "as fresh and significant as when it was discovered?two thousand years have not written a wrinkle on (it)."

Any number of books on mathematics

generally or on number theory in particular give the following version of the proof, a version that, over the years, has come to be

regarded as "authentic Euclid" (the argu ment I present is taken almost verbatim from Burton [1976, 54-55]):

Let p1 =

2, p2 =

3, p3 =

5, p4 =

7,... be the

primes in ascending order, and suppose that there is a last prime ; call it pn. Now consider the positive integer

P = PiP2P3'-Pn + !?

Since > 1, we conclude that is divis ible by some prime p. But 9 p2 , p3 , ...,

pn are the only prime numbers, so that must be equal to one of them. Since then divides evenly into both PiP2P3 ? ? ?

pn and P, we conclude that divides

evenly into

P-P1P2P3 ' "Pn

= L

But the only positive divisor of the in

teger 1 is 1 itself, contradicting the fact that > 2. Thus no finite list of primes is

complete, whence the number of primes is infinite.

Q.E.D.

This argument is indeed beautiful. It is

clever, simple, easily understood by an able

high school student, and, unfortunately, is nowhere to be found in Book IX of the Ele ments. As with the literary masterpieces cited earlier, Euclid's great theorem has en tered the folklore of mathematics in a less than-authentic form.

What theorem, then, did Euclid prove? To begin, his theorem was not to be merely a demonstration of the infinitude of primes. Rather, he stated the following :

Proposition: Prime numbers are more than

any assigned multitude of prime numbers.

That is, given any finite collection of

primes (i.e., any "assigned multitude"), Euclid promised to find a prime not con tained in this collection. Consistent with Greek mathematical taste, he framed his

argument in terms of geometric mag nitudes, but its essence is as follows (see Heath [1956, 412]):

Letting a, 6, c, e constitute his "as

signed multitude," Euclid introduced the new integer

/ = (abc ? ? ?

e) + 1

16 Mathematics Teacher

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and examined two separate cases :

Case 1: If/is prime, then / itself serves as a new prime not among the originals, since it clearly exceeds a, ?, c, ..., and

e.

Case 2: If / is not a prime, then it has a

prime divisor, say g. Euclid noted that if g = a, then g would divide evenly into both f and (abc ... ?) and, hence, g would divide evenly into f

?

(abc ... e) = 1, an impossibility. But the same impossibility arises if g = b or

g ?

c, and so on. Thus, the prime g is not one of the original multitude.

By cases 1 and 2, Euclid has shown how any finite collection of primes can be aug mented by yet another prime.

Q.E.D.

Now it is certainly true that Euclid's argument implies that the set of all primes cannot be finite. Yet, contrary to wide spread belief, Euclid did not set out to pro vide an indirect proof of the infinitude of primes but, rather, as we have seen, to dem onstrate directly how to add a new prime to any finite list. Indeed, the "popular" ver sion of Euclid's argument begins by making the dual assumptions that the "assigned

multitude" consists of consecutive primes that, for the purposes of eventual contradic tion, exhaust all the primes. Actually Euclid made neither assumption and, conse

quently, did a bit more than he is given credit for.

Worse, some students of the "popular" version incorrectly think that it has es tablished that an integer equal to one more than the product of the first consecutive primes must itself be prime, a notion dis pelled by the observation that

(2 ? 3 ? 5 ? 7 ? 11 ?

13) + 1 = 30 031 = (59)(509).

Note that because of case 2, no such con fusion arises from the Euclidean proof.

Finally, Euclid's argument differs from the "popular" version?and has an ad ditional pedagogical value?in its capabil ity of being "checked" by example. For in stance, if our initial list of primes is 2, 7, and 17, then

/?=(2?7?17) + 1 =239

is itself a prime not included among the

originals (as in case 1). However, if the ini tial list is 3, 5, and 19, then

/ = (3 ? 5 ?

19) + 1 = 286,

which has prime divisors 2, 11, and 13, none of which is among the originals (as in case

2). It may be that Euclid, like Melville, now

resides somewhere on the far shore of the River Styx quite unaware that the popular mind has altered, however slightly, his handiwork. But something should be said for approaching original genius in its orig inal form. In this regard, I recall Sir Win ston Churchill's advice to writers that "short words are best and the old words when short are best of all" (Bartlett 1980, 746) and offer the following paraphrase for mathematicians: "Short proofs are best and the old proofs when short are best of all."

REFERENCES

Armour, Richard. The Classics Reclassified. New York: McGraw-Hill Book Co., I960.

Bartlett, John. Bartlett's Familiar Quotations. 15th ed. Boston: Little, Brown & Co., 1980.

Burton, David. Elementary Number Theory. Boston: Allyn & Bacon, 1976.

Hardy, G. H. A Mathematician's Apology. Cambridge: Cambridge University Press, 1969.

Heath, T. L. The Thirteen Books of Euclid's Elements. Vol. 2. Reprint. New York: Dover Publications, 1956.

Shakespeare, William. The Tragedy of Hamlet, Prince of Denmark. New York: New American Library, 1963. m

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