平成 30 年度 東北大学 大学院理学研究科 数学専攻 入学試験問題

平成 30 年度 東北大学 大学院理学研究科 数学専攻 入学試験問題

英語

平成

29

8

22

17

注意事項

1)

2)

2

3)

4)

2

3

1

1

^{W. Dunham,} Euler: The Master of Us All, The Mathematical Association

of America (1999)

(1)–(3)

Over two millennia ago Euclid asked, and answered, a basic question about the primes in what is one of the greatest proofs ever devised. As Proposition 20 of Book IX of the Elements, he showed that no finite collection of primes—no matter how vast—can possibly include them all. Although his argument has been reproduced countless times, it always warrants a quick reprise.

Theorem. No finite collection of primes includes them all.

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Euclid left the matter there, but prime numbers—their characteristics, their structure, their distribution—have been among the most studied of mathematical objects,

exhibiting a fascination as endless as the primes themselves.

For instance, consider this dichotomy among the odd primes: each is either of the form 4k + 1 or 4k − 1. That is, an odd prime (indeed, any odd number) is either one more or one less than a multiple of 4.

Initially we might try to assess the relative abundance of the two types of primes.

Among the first hundred numbers, the primes in each family are:

4k + 1 : 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97 4k − 1 : 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83.

Among the next hundred, we have:

4k + 1 : 101, 109, 113, 137, 149, 157, 173, 181, 193, 197 4k − 1 : 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199.

Patterns here are not conspicuous, but from these scanty data one is tempted to pro- pose that, in any string of whole numbers starting from 1, there are slightly more 4k − 1 primes than their 4k +1 counterparts. In other words, one might surmise that, no matter the size of n, the 4k − 1 primes are in the majority among the numbers 1, 2, 3, . . . , n.

This conjecture

is false. The 4k + 1 primes eventually overtake the 4k − 1 primes, but (strangely) this does not happen until we consider the very long string 1, 2, 3, . . . , 26861.

Only then does the balance finally tip in favor of the 4k + 1 type. Soon thereafter it tips back—and then back again. The twentieth century mathematician J. E. Littlewood (1885–1977) showed that the majority status changes hands infinitely often as we march through the positive integers. It is like a two-horse race in which neither thoroughbred can maintain the lead.

Long before Littlewood, mathematicians had raised the question of the overall abun- dance of the two types of primes. Because there are infinitely many primes, it is clear that at least one of the two families must itself be infinite. With an argument modeled upon Euclid’s, we prove:

Theorem. There are infinitey many 4k − 1 primes.

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2

    著作権上の制約により公開していません。 

And what of the abundance of the 4k + 1 primes? The preceding theorem does not allow us to conclude anything in this direction. True, we know there are infinitely many primes altogether, and infinitely many of them are of the 4k − 1 variety, but this is logically insufficient to determine the finitude or infinitude of the other type.

The fact that there are infinitely many of the 4k + 1 primes—although true—turned out to be much more difficult to prove.

(1)

(a)

(2)

(b)

This conjecture

. (3)

(c)

2

(1)

{ a

}

M

n

a

≤ M

上に有界であるという．

(2)

n × n

,

.

, n × n

.

3

    著作権上の制約により公開していません。