오일러의 convenient number ( Idoneal number)

간단한 소개

이차형식에 대한 오일러의 연구에서 발견
Numeri Idonei
현재까지 알려진 목록
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848
이 목록이 완전한 목록인지는 아직 미해결 문제. 있으면 단 하나가 더 있을 수 있음이 증명되었음

오일러의 정의

자연수 \(m\)이 다음 조건을 만족시킬 때, convenient 라고 한다

홀수 \(n > 1\) 이 이차형식\(x^2+my^2\)에 의하여 단 한가지 방법으로 표현되면, (\(x,y\)는 음이 아닌 정수이고 \((x, my) = 1\)), \(n\)은 소수이다

오일러의 판정법

증명되지 않은 오일러의 판정법

A number \(m\in \mathbb{N}\) is convenient

if and only if

every natural number \(n\) of the form \(n = m + x^2 <4m\) with \(x\in \mathbb{N}\), \((x,m) = 1\) is necessarily of one of the four forms \(n = p\), \(n = 2p\), \(n = p^2\), \(n = 2^s\) where \(p\) is an odd prime number and \(s\in \mathbb{N}\)

오일러의 판정법 사용예

\(m=13\)
\(13 + 1^2 = 14 = 2p\)
\(13 + 2^2 = 17 = p\)
\(13 + 3^2 = 22 = 2p\)
\(13 + 4^2 = 29 = p\)
\(13 + 5^2 = 38 = 2p\)
\(13 + 6^2 = 49 = p^2\)
따라서 \(m=13\) 은 convenient
\(m=15\)
\(15 + 1^2 = 16 = 2^4\)
\(15 + 2^2 = 19 = p\)
\(15 + 4^2 = 31 = p\)
따라서 \(m=15\) 는 convenient

\(m=14\)
\(14 + 1^2 = 15 = 3 \cdot 5\)
따라서 \(m=14\) 는 convenient 가 아님

오일러가 발견한 성질들

If m is convenient and \(m = t^2\), then \(t=1,2,3,4,5\).
If m is convenient and \(m \equiv 3 \pmod 4\), then 4m is convenient.
- 예) m= 3,7,15, 4m=12, 28, 60
If m is convenient and \(m \equiv 4 \pmod 8\), then 4m is convenient.
- 예) m= 4,12,28, 60 , 4m = 16, 48, 112, 240
If \(k^2 m\) is convenient, then m is convenient.
- \(k^2 m\)= 4, 8, 9, 12, 16, 18, 24, 25, 28, 40, 45, 48, 60, 72, 88, 112, 120, 168, 232, 240, 280, 312, 408, 520, 760, 840, 1320, 1848
If m is convenient and \(m \equiv 2 \pmod 3\) , then 9m is convenient.
- m=2,5,8 , 4m =
If m > 1 is convenient and \(m \equiv 1 \pmod 4\) , then 4m is not convenient.
If m is convenient and \(m \equiv 2 \pmod 4\), then 4m is convenient.
If m is convenient and \(m \equiv 8 \pmod {16}\), then 4m is not convenient.
If m is convenient and \(m \equiv 16 \pmod {32}\), then 4m is not convenient.
If m is convenient and \(m + a^2 = p^2 < 4m\) for a prime p, then 4m is not convenient.

가우스의 판정법

(a) A number \(m\in \mathbb{N}\) is convenient if and only if every genus of properly primitive integral binary quadratic forms of determinant d = - m contains precisely one proper class of properly primitive forms;
or alternatively,
(b) A number \(m\in \mathbb{N}\) is convenient if and only if every proper class of properly primitive integral binary quadratic forms with determinant d = -m is a proper ambiguous class of properly primitive forms.

Grube의 판정법 1

A number \(m\in \mathbb{N}\) is convenient if and only if every natural number n of the form
\(n = m + x^2\)with \(x\in \mathbb{N}\) and \(x < \sqrt{\frac{m}{3}}\) admits no factorizations \(n = rs\) with \(s \geq r \geq 2x\), \(r, s \in \mathbb{N}\) except those of the form \(r=s\) or \(r=2x\).

Grube의 판정법 사용예

\(m=48\)
\(48 + 1^2 = 49 = 7\cdot 7 : r = s\)
\(48 + 2^2 = 52 = 4\cdot 13 : r = 2x\)
\(48 + 3^2 = 57\)
\(48 + 4^2 = 64 = 8\cdot 8 : r = s\)
따라서 \(m=48\) 은 convenient
\(m=60\)
\(60 + 1^2 = 61\)
\(60 + 2^2 = 64 = 8\cdot 8 : r = s\)
\(60 + 3^2 = 69\)
\(60 + 4^2 = 76\)
따라서 \(m=60\) 은 convenient
\(m=11\)
\(11+1^2=12=3\cdot 4\)
따라서 \(m=11\) 은 convenient가 아님

Grube의 판정법 2

Suppose \(m\in \mathbb{N}\) is not divisible by a square and suppose \(m\neq 3,7,15\)
Then m is convenient if and only if every natural number n of the form

\(n = m + x^2\)with \(x\in \mathbb{N}\) and \(x < \sqrt{\frac{m}{3}}\)
is also of the form
\(n = tp\), \(n = 2tp\) or \(n = p^2\)
where t is a divisor of m, and p is an odd prime number.

Grube의 판정법 2 사용예

\(m=30\)
\(30 + 1^2 = 31 = p\)
\(30 + 2^2 = 34 = 2\cdot 17 = 2p\)
\(30 + 3^2 = 39 = 3\cdot 13 = tp\)
따라서 \(m=30\) 은 convenient

또다른 성질들

Let \(m\in \mathbb{N}\) . Then all prime numbers p of the form \(p = x^2 + my^2\)with \(x,y \in \mathbb{N}\) can be characterized by congruence conditions with respect to a single modulus f if and only if m is convenient.

class number 에 따른 분류

\(h(-4n)\)	n's with one class per genus
1	1,2,3,4,7
2	5,6,8,9,10,12,13,15,16,18,22,25,28,37,58
4	21,24,30,33,40,42,45,48,57,60,70,72,78,85,88,93,102,112,130,133,177,190,232,253
8	105,120,165,168,210,240,273,280,312,330,345,357,385,408,462,520,760
16	840,1320,1365,1848

메모

Baltes, H. P. and Hill E. R.: Spectra of Finite Systems. Bibliographisches Institut, Z~irich, 1976
Chowla, S.: An Extension of Heilbronn's Class Number Theorem. Quarterly J. Math. (Oxford) 5 (1934), 304-307
Euler, L.: Opera Omnia. Series Prima. Teubner, Leipzig, 1911-
Fermat, P.: Oeuvres. Tome 2, 212-217, Gauthier-Villars, Paris, 1894
Frei, G.: On the Development of the Genus of Quadratic Forms. Ann. Sci. Math. Qu6bec 3 (1979), 5-62
Frei, G.: Les nombres convenables de Leonhard Euler.(To appear)
Gauss, C. F.: Disquisitiones arithmeticae. Leipzig, 1801(or: Untersuchungen tiber h6here Mathematik. Herausgegeben von H. Maser, Springer, Berlin, 1889)
Grube, F.: Ueber einige Eulersche S/itze aus der Theorie der quadratischen Formen. Zeitschrift f~ir Mathematik und Physik 19 (1874), 492-519
Lagrange, J.-L.: Recherches d'arithm6tique, 1773 et 1775. Oeuvres, Tome 3, Gauthier-Villars, Paris, 1867
Steinig, J.: On Euler's Idoenal Numbers. Elemente der Mathematik 21 (1966), 73-88

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