"오일러의 convenient number ( Idoneal number)"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 28개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5>간단한 소개</h5>
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==개요==
  
 
* 이차형식에 대한 오일러의 연구에서 발견
 
* 이차형식에 대한 오일러의 연구에서 발견
 
* Numeri Idonei
 
* Numeri Idonei
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*  현재까지 알려진 목록<br>
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*  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
 
** 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
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* 이 목록이 완전한 목록인지는 아직 미해결 문제. 있으면 단 하나가 더 있을 수 있음이 증명되었음
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* '''[Chowla1954] '''는 이 목록이 유한임을 증명
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* 이 65개의 목록이 완전한 목록인지는 아직 미해결.
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* 복소이차수체의 class group의 exponent가 2 이하인 경우를 통해 이해할 수 있음
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* 이러한 성질을 갖는 복소이차수체는 지금까지 알려진 경우 외에 많아야 하나가 더 있을 수 있음이 증명, [[디리클레 L-함수]]에 대한 일반화된 리만가설이 성립할 경우, 문제 해결 '''[Weinberger73]'''
  
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<h5>오일러의 정의</h5>
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==오일러의 정의==
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* 자연수 <math>m</math>이 다음 조건을 만족시킬 때, convenient 라고 한다 :
  
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* 자연수 <math>m</math>이 다음 조건을 만족시킬 때, convenient 라고 한다
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홀수 <math>n > 1</math> 이 이차형식<math>x^2+my^2</math>에 의하여 단 한가지 방법으로 표현되면, (<math>x,y</math>는 음이 아닌 정수이고 <math>(x, my) = 1</math>), <math>n</math>은 소수이다
  
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홀수 <math>n > 1</math> 이 이차형식<math>x^2+my^2</math>에 의하여 단 한가지 방법으로 표현되면, (<math>x,y</math>는 음이 아닌 정수이고 <math>(x, my) = 1</math>), <math>n</math>은 소수이다
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* 큰 소수를 찾아내는데 활용:<math>18518809=197^2+1848\cdot 100^2</math>
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* 1848이 convenient 임을 이용하여 18518809라는 당시로서는 큰 소수를 찾아냄
  
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<h5>오일러의 판정법</h5>
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==오일러의 판정법==
  
 
* 증명되지 않은 오일러의 판정법
 
* 증명되지 않은 오일러의 판정법
  
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A number <math>m\in \mathbb{N}</math> is convenient
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A number <math>m\in \mathbb{N}</math> is convenient
  
 
if and only if
 
if and only if
  
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every natural number <math>n</math> of the form <math>n = m + x^2 <4m</math> with <math>x\in \mathbb{N}</math>, <math>(x,m) = 1</math> is necessarily of one of the four forms <math>n = p</math>, <math>n = 2p</math>, <math>n = p^2</math>, <math>n = 2^s</math> where <math>p</math> is an odd prime number and <math>s\in \mathbb{N}</math>
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every natural number <math>n</math> of the form <math>n = m + x^2 <4m</math> with <math>x\in \mathbb{N}</math>, <math>(x,m) = 1</math> is necessarily of one of the four forms <math>n = p</math>, <math>n = 2p</math>, <math>n = p^2</math>, <math>n = 2^s</math> where <math>p</math> is an odd prime number and <math>s\in \mathbb{N}</math>
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<h5>오일러의 판정법 사용예</h5>
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==오일러의 판정법 사용예==
  
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* <math>m=13</math><br><math>13 + 1^2 = 14 = 2p</math><br><math>13 + 2^2 = 17 = p</math><br><math>13 + 3^2 = 22 = 2p</math><br><math>13 + 4^2 = 29 = p</math><br><math>13 + 5^2 = 38 = 2p</math><br><math>13 + 6^2 = 49 = p^2</math><br> 따라서 <math>m=13</math> 은 convenient<br>
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* <math>m=13</math>:<math>13 + 1^2 = 14 = 2p</math>:<math>13 + 2^2 = 17 = p</math>:<math>13 + 3^2 = 22 = 2p</math>:<math>13 + 4^2 = 29 = p</math>:<math>13 + 5^2 = 38 = 2p</math>:<math>13 + 6^2 = 49 = p^2</math> 따라서 <math>m=13</math> 은 convenient
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* <math>m=15</math><br><math>15 + 1^2 = 16 = 2^4</math><br><math>15 + 2^2 = 19 = p</math><br><math>15 + 4^2 = 31 = p</math><br> 따라서 <math>m=15</math> 는 convenient<br>
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* <math>m=15</math>:<math>15 + 1^2 = 16 = 2^4</math>:<math>15 + 2^2 = 19 = p</math>:<math>15 + 4^2 = 31 = p</math> 따라서 <math>m=15</math> 는 convenient
  
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* <math>m=14</math><br><math>14 + 1^2 = 15 = 3 \cdot 5</math><br> 따라서 <math>m=14</math> 는 convenient 가 아님<br>
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* <math>m=14</math>:<math>14 + 1^2 = 15 = 3 \cdot 5</math> 따라서 <math>m=14</math> 는 convenient 가 아님
  
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<h5>오일러가 발견한 성질들</h5>
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==오일러가 발견한 성질들==
  
 
* If m is convenient and <math>m = t^2</math>, then <math>t=1,2,3,4,5</math>.
 
* If m is convenient and <math>m = t^2</math>, then <math>t=1,2,3,4,5</math>.
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*  If m is convenient and <math>m \equiv 3 \pmod 4</math>, then 4m is convenient.<br>
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*  If m is convenient and <math>m \equiv 3 \pmod 4</math>, then 4m is convenient.
 
** 예) m= 3,7,15, 4m=12, 28, 60
 
** 예) m= 3,7,15, 4m=12, 28, 60
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*  If m is convenient and <math>m \equiv 4 \pmod 8</math>, then 4m is convenient.<br>
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*  If m is convenient and <math>m \equiv 4 \pmod 8</math>, then 4m is convenient.
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** 예) m= 4,12,28, 60  , 4m = 16, 48, 112, 240
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** 예) m= 4,12,28, 60  , 4m = 16, 48, 112, 240
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*  If <math>k^2 m</math> is convenient, then m is convenient.<br>
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*  If <math>k^2 m</math> is convenient, then m is convenient.
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** {{4}, {8}, {9}, {12}, {16}, {18}, {24}, {25}, {28}, {40}, {45}, {48}, \<br> {60}, {72}, {88}, {112}, {120}, {168}, {232}, {240}, {280}, {312}, \<br> {408}, {520}, {760}, {840}, {1320}, {1848}}
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** <math>k^2 m</math>= 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
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* If m is convenient and <math>m \equiv 2 \pmod 3</math> , then 9m is convenient.
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* If m is convenient and <math>m \equiv 2 \pmod 3</math> , then 9m is convenient.
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* If m > 1 is convenient and <math>m \equiv 1 \pmod 4</math> , then 4m is not convenient.
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** m=2,5,8 , 4m = 18, 45, 72
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* If m is convenient and <math>m \equiv 2 \pmod 4</math>, then 4m is convenient.
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*  If m > 1 is convenient and <math>m \equiv 1 \pmod 4</math> , then 4m is not convenient.
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* If m is convenient and <math>m \equiv 8 \pmod {16}</math>, then 4m is not convenient.
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** m = 5, 9, 13, 21, 25, 33, 37, 45, 57, 85, 93, 105, 133, 165, 177, 253, 273, 345, 357, 385, 1365
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* If m is convenient and <math>m \equiv 16 \pmod {32}</math>, then 4m is not convenient.
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** 4m = 20, 36, 52, 84, 100, 132, 148, 180, 228, 340, 372, 420, 532, 660, 708, 1012, 1092, 1380, 1428, 1540, 5460
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* If m is convenient and <math>m + a^2 = p^2 < 4m</math> for a prime p, then 4m is not convenient.
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*  If m is convenient and <math>m \equiv 2 \pmod 4</math>, then 4m is convenient.
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** m = 2, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462
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** 4m = 8, 24, 40, 72, 88, 120, 168, 232, 280, 312, 408, 520, 760, 840, 1320, 1848
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*  If m is convenient and <math>m \equiv 8 \pmod {16}</math>, then 4m is not convenient.
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** m = 8, 24, 40, 72, 88, 120, 168, 232, 280, 312, 408, 520, 760, 840, 1320, 1848
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** 4m = 32, 96, 160, 288, 352, 480, 672, 928, 1120, 1248, 1632, 2080, 3040, 3360, 5280, 7392
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*  If m is convenient and <math>m \equiv 16 \pmod {32}</math>, then 4m is not convenient.
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** m = 16, 48, 112, 240
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** 4m = 64, 192, 448, 960
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* If m is convenient and <math>m + x^2 = p^2 < 4m</math> for a prime p, then 4m is not convenient.
  
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<h5>가우스의 판정법</h5>
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==가우스의 판정법==
  
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(a) A number <math>m\in \mathbb{N}</math> 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;<br> or alternatively,<br> (b) A number <math>m\in \mathbb{N}</math> 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.
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(a) A number <math>m\in \mathbb{N}</math> is convenient if and only if every genus of properly primitive integral binary quadratic forms of discriminant <math>\Delta = -4m</math> contains precisely one proper class of properly primitive forms; or alternatively, (b) A number <math>m\in \mathbb{N}</math> is convenient if and only if every proper class of properly primitive integral binary quadratic forms discriminant <math>\Delta = -4m</math> is a proper ambiguous class of properly primitive forms.
  
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<h5>Grube의 판정법 1</h5>
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==Grube의 판정법 1==
  
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A number <math>m\in \mathbb{N}</math>  is convenient if and only if every natural number n of the form<br><math>n = m + x^2</math>with <math>x\in \mathbb{N}</math> and <math>x < \sqrt{\frac{m}{3}}</math> admits no factorizations <math>n = rs</math> with <math>s \geq r \geq 2x</math>, <math>r, s \in \mathbb{N}</math> except those of the form <math>r=s</math> or <math>r=2x</math>.
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A number <math>m\in \mathbb{N}</math> is convenient if and only if every natural number n of the form:<math>n = m + x^2</math>with <math>x\in \mathbb{N}</math> and <math>x < \sqrt{\frac{m}{3}}</math> admits no factorizations <math>n = rs</math> with <math>s \geq r \geq 2x</math>, <math>r, s \in \mathbb{N}</math> except those of the form <math>r=s</math> or <math>r=2x</math>.
  
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<h5> </h5>
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==Grube의 판정법 1 사용예==
  
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<h5>Grube의 판정법 사용예</h5>
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* <math>m=48</math>:<math>48 + 1^2 = 49 = 7\cdot 7 : r = s</math>:<math>48 + 2^2 = 52 = 4\cdot 13 : r = 2x</math>:<math>48 + 3^2 = 57</math>:<math>48 + 4^2 = 64 = 8\cdot 8 : r = s</math> 따라서 <math>m=48</math> 은 convenient
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* <math>m=60</math>:<math>60 + 1^2 = 61</math>:<math>60 + 2^2 = 64 = 8\cdot 8 : r = s</math>:<math>60 + 3^2 = 69</math>:<math>60 + 4^2 = 76</math> 따라서 <math>m=60</math> 은 convenient
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* <math>m=11</math>:<math>11+1^2=12=3\cdot 4</math> 따라서 <math>m=11</math> 은 convenient가 아님 
  
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* <math>m=48</math><br><math>48 + 1^2 = 49 = 7\cdot 7 : r = s</math><br><math>48 + 2^2 = 52 = 4\cdot 13 : r = 2x</math><br><math>48 + 3^2 = 57</math><br><math>48 + 4^2 = 64 = 8\cdot 8 : r = s</math><br> 따라서 <math>m=48</math> 은 convenient<br>
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* <math>m=60</math><br><math>60 + 1^2 = 61</math><br><math>60 + 2^2 = 64 = 8\cdot 8 : r = s</math><br><math>60 + 3^2 = 69</math><br><math>60 + 4^2 = 76</math><br> 따라서 <math>m=60</math> 은 convenient<br>
 
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* <math>m=11</math><br><math>11+1^2=12=3\cdot 4</math><br> 따라서 <math>m=11</math> 은 convenient가 아님<br>  <br>
 
  
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==Grube의 판정법 2==
  
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<h5>Grube의 판정법 2</h5>
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Suppose <math>m\in \mathbb{N}</math> is not divisible by a square and suppose <math>m\neq 3,7,15</math> Then m is convenient if and only if every natural number n of the form
  
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Suppose <math>m\in \mathbb{N}</math> is not divisible by a square and suppose <math>m\neq 3,7,15</math><br> Then m is convenient if and only if every natural number n of the form
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<math>n = m + x^2</math>with <math>x\in \mathbb{N}</math> and <math>x < \sqrt{\frac{m}{3}}</math> is also of the form:<math>n = tp</math>, <math>n = 2tp</math> or <math>n = p^2</math> where t is a divisor of m, and p is an odd prime number.
  
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<math>n = m + x^2</math>with <math>x\in \mathbb{N}</math> and <math>x < \sqrt{\frac{m}{3}}</math><br> is also of the form<br><math>n = tp</math>, <math>n = 2tp</math> or <math>n = p^2</math><br> where t is a divisor of m, and p is an odd prime number.
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==Grube의 판정법 2 사용예==
  
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<h5>Grube의 판정법 2 사용예</h5>
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* <math>m=30</math>:<math>30 + 1^2 = 31 = p</math>:<math>30 + 2^2 = 34 = 2\cdot 17 = 2p</math>:<math>30 + 3^2 = 39 = 3\cdot 13 = tp</math> 따라서 <math>m=30</math> 은 convenient
  
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* <math>m=30</math><br><math>30 + 1^2 = 31 = p</math><br><math>30 + 2^2 = 34 = 2\cdot 17 = 2p</math><br><math>30 + 3^2 = 39 = 3\cdot 13 = tp</math><br> 따라서 <math>m=30</math> 은 convenient<br>
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==또다른 성질들==
  
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<h5>또다른 성질들</h5>
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Let <math>m\in \mathbb{N}</math> .
  
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Let <math>m\in \mathbb{N}</math> . Then all prime numbers p of the form <math>p = x^2 + my^2</math>with <math>x,y \in \mathbb{N}</math> can be characterized by congruence conditions with respect to a single modulus f if and only if m is convenient.
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Then all prime numbers p of the form <math>p = x^2 + my^2</math>with <math>x,y \in \mathbb{N}</math> can be characterized by congruence conditions with respect to a single modulus f
  
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if and only if
  
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<h5>class number 에 따른 분류</h5>
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m is convenient.
  
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==class number 에 따른 분류==
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{| class="dataTable2" style=""
 
{| class="dataTable2" style=""
124번째 줄: 144번째 줄:
 
| <math>h(-4n)</math>
 
| <math>h(-4n)</math>
 
| n's with one class per genus
 
| n's with one class per genus
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| number of such convenient numbers
 
|-
 
|-
 
| 1
 
| 1
 
| 1,2,3,4,7
 
| 1,2,3,4,7
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| 5
 
|-
 
|-
 
| 2
 
| 2
 
| 5,6,8,9,10,12,13,15,16,18,22,25,28,37,58
 
| 5,6,8,9,10,12,13,15,16,18,22,25,28,37,58
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| 15
 
|-
 
|-
 
| 4
 
| 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
 
| 21,24,30,33,40,42,45,48,57,60,70,72,78,85,88,93,102,112,130,133,177,190,232,253
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| 24
 
|-
 
|-
 
| 8
 
| 8
 
| 105,120,165,168,210,240,273,280,312,330,345,357,385,408,462,520,760
 
| 105,120,165,168,210,240,273,280,312,330,345,357,385,408,462,520,760
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| 17
 
|-
 
|-
 
| 16
 
| 16
 
| 840,1320,1365,1848
 
| 840,1320,1365,1848
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| 4
 
|}
 
|}
  
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<h5>메모</h5>
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==convenient number에 대한 이차형식의 목록==
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 +
n=1,{x^2+y^2} n=2,{x^2+2 y^2} n=3,{x^2+3 y^2} n=4,{x^2+4 y^2} n=5,{x^2+5 y^2,2 x^2+2 x y+3 y^2} n=6,{x^2+6 y^2,2 x^2+3 y^2} n=7,{x^2+7 y^2} n=8,{x^2+8 y^2,3 x^2+2 x y+3 y^2} n=9,{x^2+9 y^2,2 x^2+2 x y+5 y^2} n=10,{x^2+10 y^2,2 x^2+5 y^2} n=12,{x^2+12 y^2,3 x^2+4 y^2} n=13,{x^2+13 y^2,2 x^2+2 x y+7 y^2} n=15,{x^2+15 y^2,3 x^2+5 y^2} n=16,{x^2+16 y^2,4 x^2+4 x y+5 y^2} n=18,{x^2+18 y^2,2 x^2+9 y^2} n=21,{x^2+21 y^2,2 x^2+2 x y+11 y^2,3 x^2+7 y^2,5 x^2+4 x y+5 y^2} n=22,{x^2+22 y^2,2 x^2+11 y^2} n=24,{x^2+24 y^2,3 x^2+8 y^2,4 x^2+4 x y+7 y^2,5 x^2+2 x y+5 y^2} n=25,{x^2+25 y^2,2 x^2+2 x y+13 y^2} n=28,{x^2+28 y^2,4 x^2+7 y^2} n=30,{x^2+30 y^2,2 x^2+15 y^2,3 x^2+10 y^2,5 x^2+6 y^2} n=33,{x^2+33 y^2,2 x^2+2 x y+17 y^2,3 x^2+11 y^2,6 x^2+6 x y+7 y^2} n=37,{x^2+37 y^2,2 x^2+2 x y+19 y^2} n=40,{x^2+40 y^2,4 x^2+4 x y+11 y^2,5 x^2+8 y^2,7 x^2+6 x y+7 y^2} n=42,{x^2+42 y^2,2 x^2+21 y^2,3 x^2+14 y^2,6 x^2+7 y^2} n=45,{x^2+45 y^2,2 x^2+2 x y+23 y^2,5 x^2+9 y^2,7 x^2+4 x y+7 y^2} n=48,{x^2+48 y^2,3 x^2+16 y^2,4 x^2+4 x y+13 y^2,7 x^2+2 x y+7 y^2} n=57,{x^2+57 y^2,2 x^2+2 x y+29 y^2,3 x^2+19 y^2,6 x^2+6 x y+11 y^2} n=58,{x^2+58 y^2,2 x^2+29 y^2} n=60,{x^2+60 y^2,3 x^2+20 y^2,4 x^2+15 y^2,5 x^2+12 y^2} n=70,{x^2+70 y^2,2 x^2+35 y^2,5 x^2+14 y^2,7 x^2+10 y^2} n=72,{x^2+72 y^2,4 x^2+4 x y+19 y^2,8 x^2+9 y^2,8 x^2+8 x y+11 y^2} n=78,{x^2+78 y^2,2 x^2+39 y^2,3 x^2+26 y^2,6 x^2+13 y^2} n=85,{x^2+85 y^2,2 x^2+2 x y+43 y^2,5 x^2+17 y^2,10 x^2+10 x y+11 y^2} n=88,{x^2+88 y^2,4 x^2+4 x y+23 y^2,8 x^2+11 y^2,8 x^2+8 x y+13 y^2} n=93,{x^2+93 y^2,2 x^2+2 x y+47 y^2,3 x^2+31 y^2,6 x^2+6 x y+17 y^2} n=102,{x^2+102 y^2,2 x^2+51 y^2,3 x^2+34 y^2,6 x^2+17 y^2} n=105,{x^2+105 y^2,2 x^2+2 x y+53 y^2,3 x^2+35 y^2,5 x^2+21 y^2,6 x^2+6 x y+19 y^2,7 x^2+15 y^2,10 x^2+10 x y+13 y^2,11 x^2+8 x y+11 y^2} n=112,{x^2+112 y^2,4 x^2+4 x y+29 y^2,7 x^2+16 y^2,11 x^2+6 x y+11 y^2} n=120,{x^2+120 y^2,3 x^2+40 y^2,4 x^2+4 x y+31 y^2,5 x^2+24 y^2,8 x^2+15 y^2,8 x^2+8 x y+17 y^2,11 x^2+2 x y+11 y^2,12 x^2+12 x y+13 y^2} n=130,{x^2+130 y^2,2 x^2+65 y^2,5 x^2+26 y^2,10 x^2+13 y^2} n=133,{x^2+133 y^2,2 x^2+2 x y+67 y^2,7 x^2+19 y^2,13 x^2+12 x y+13 y^2} n=165,{x^2+165 y^2,2 x^2+2 x y+83 y^2,3 x^2+55 y^2,5 x^2+33 y^2,6 x^2+6 x y+29 y^2,10 x^2+10 x y+19 y^2,11 x^2+15 y^2,13 x^2+4 x y+13 y^2} n=168,{x^2+168 y^2,3 x^2+56 y^2,4 x^2+4 x y+43 y^2,7 x^2+24 y^2,8 x^2+21 y^2,8 x^2+8 x y+23 y^2,12 x^2+12 x y+17 y^2,13 x^2+2 x y+13 y^2} n=177,{x^2+177 y^2,2 x^2+2 x y+89 y^2,3 x^2+59 y^2,6 x^2+6 x y+31 y^2} n=190,{x^2+190 y^2,2 x^2+95 y^2,5 x^2+38 y^2,10 x^2+19 y^2} n=210,{x^2+210 y^2,2 x^2+105 y^2,3 x^2+70 y^2,5 x^2+42 y^2,6 x^2+35 y^2,7 x^2+30 y^2,10 x^2+21 y^2,14 x^2+15 y^2} n=232,{x^2+232 y^2,4 x^2+4 x y+59 y^2,8 x^2+29 y^2,8 x^2+8 x y+31 y^2} n=240,{x^2+240 y^2,3 x^2+80 y^2,4 x^2+4 x y+61 y^2,5 x^2+48 y^2,12 x^2+12 x y+23 y^2,15 x^2+16 y^2,16 x^2+16 x y+19 y^2,17 x^2+14 x y+17 y^2} n=253,{x^2+253 y^2,2 x^2+2 x y+127 y^2,11 x^2+23 y^2,17 x^2+12 x y+17 y^2} n=273,{x^2+273 y^2,2 x^2+2 x y+137 y^2,3 x^2+91 y^2,6 x^2+6 x y+47 y^2,7 x^2+39 y^2,13 x^2+21 y^2,14 x^2+14 x y+23 y^2,17 x^2+8 x y+17 y^2} n=280,{x^2+280 y^2,4 x^2+4 x y+71 y^2,5 x^2+56 y^2,7 x^2+40 y^2,8 x^2+35 y^2,8 x^2+8 x y+37 y^2,17 x^2+6 x y+17 y^2,19 x^2+18 x y+19 y^2} n=312,{x^2+312 y^2,3 x^2+104 y^2,4 x^2+4 x y+79 y^2,8 x^2+39 y^2,8 x^2+8 x y+41 y^2,12 x^2+12 x y+29 y^2,13 x^2+24 y^2,19 x^2+14 x y+19 y^2} n=330,{x^2+330 y^2,2 x^2+165 y^2,3 x^2+110 y^2,5 x^2+66 y^2,6 x^2+55 y^2,10 x^2+33 y^2,11 x^2+30 y^2,15 x^2+22 y^2} n=345,{x^2+345 y^2,2 x^2+2 x y+173 y^2,3 x^2+115 y^2,5 x^2+69 y^2,6 x^2+6 x y+59 y^2,10 x^2+10 x y+37 y^2,15 x^2+23 y^2,19 x^2+8 x y+19 y^2} n=357,{x^2+357 y^2,2 x^2+2 x y+179 y^2,3 x^2+119 y^2,6 x^2+6 x y+61 y^2,7 x^2+51 y^2,14 x^2+14 x y+29 y^2,17 x^2+21 y^2,19 x^2+4 x y+19 y^2} n=385,{x^2+385 y^2,2 x^2+2 x y+193 y^2,5 x^2+77 y^2,7 x^2+55 y^2,10 x^2+10 x y+41 y^2,11 x^2+35 y^2,14 x^2+14 x y+31 y^2,22 x^2+22 x y+23 y^2} n=408,{x^2+408 y^2,3 x^2+136 y^2,4 x^2+4 x y+103 y^2,8 x^2+51 y^2,8 x^2+8 x y+53 y^2,12 x^2+12 x y+37 y^2,17 x^2+24 y^2,23 x^2+22 x y+23 y^2} n=462,{x^2+462 y^2,2 x^2+231 y^2,3 x^2+154 y^2,6 x^2+77 y^2,7 x^2+66 y^2,11 x^2+42 y^2,14 x^2+33 y^2,21 x^2+22 y^2} n=520,{x^2+520 y^2,4 x^2+4 x y+131 y^2,5 x^2+104 y^2,8 x^2+65 y^2,8 x^2+8 x y+67 y^2,13 x^2+40 y^2,20 x^2+20 x y+31 y^2,23 x^2+6 x y+23 y^2} n=760,{x^2+760 y^2,4 x^2+4 x y+191 y^2,5 x^2+152 y^2,8 x^2+95 y^2,8 x^2+8 x y+97 y^2,19 x^2+40 y^2,20 x^2+20 x y+43 y^2,29 x^2+18 x y+29 y^2} n=840,{x^2+840 y^2,3 x^2+280 y^2,4 x^2+4 x y+211 y^2,5 x^2+168 y^2,7 x^2+120 y^2,8 x^2+105 y^2,8 x^2+8 x y+107 y^2,12 x^2+12 x y+73 y^2,15 x^2+56 y^2,20 x^2+20 x y+47 y^2,21 x^2+40 y^2,24 x^2+35 y^2,24 x^2+24 x y+41 y^2,28 x^2+28 x y+37 y^2,29 x^2+2 x y+29 y^2,31 x^2+22 x y+31 y^2} n=1320,{x^2+1320 y^2,3 x^2+440 y^2,4 x^2+4 x y+331 y^2,5 x^2+264 y^2,8 x^2+165 y^2,8 x^2+8 x y+167 y^2,11 x^2+120 y^2,12 x^2+12 x y+113 y^2,15 x^2+88 y^2,20 x^2+20 x y+71 y^2,24 x^2+55 y^2,24 x^2+24 x y+61 y^2,33 x^2+40 y^2,37 x^2+14 x y+37 y^2,40 x^2+40 x y+43 y^2,41 x^2+38 x y+41 y^2} n=1365,{x^2+1365 y^2,2 x^2+2 x y+683 y^2,3 x^2+455 y^2,5 x^2+273 y^2,6 x^2+6 x y+229 y^2,7 x^2+195 y^2,10 x^2+10 x y+139 y^2,13 x^2+105 y^2,14 x^2+14 x y+101 y^2,15 x^2+91 y^2,21 x^2+65 y^2,26 x^2+26 x y+59 y^2,30 x^2+30 x y+53 y^2,35 x^2+39 y^2,37 x^2+4 x y+37 y^2,42 x^2+42 x y+43 y^2} n=1848,{x^2+1848 y^2,3 x^2+616 y^2,4 x^2+4 x y+463 y^2,7 x^2+264 y^2,8 x^2+231 y^2,8 x^2+8 x y+233 y^2,11 x^2+168 y^2,12 x^2+12 x y+157 y^2,21 x^2+88 y^2,24 x^2+77 y^2,24 x^2+24 x y+83 y^2,28 x^2+28 x y+73 y^2,33 x^2+56 y^2,43 x^2+2 x y+43 y^2,44 x^2+44 x y+53 y^2,47 x^2+38 x y+47 y^2}
 +
 
 +
 +
 
 +
==메모==
  
 
* Baltes, H. P. and Hill E. R.: Spectra of Finite Systems. Bibliographisches Institut, Z~irich, 1976
 
* Baltes, H. P. and Hill E. R.: Spectra of Finite Systems. Bibliographisches Institut, Z~irich, 1976
156번째 줄: 190번째 줄:
 
* Steinig, J.: On Euler's Idoenal Numbers. Elemente der Mathematik 21 (1966), 73-88
 
* Steinig, J.: On Euler's Idoenal Numbers. Elemente der Mathematik 21 (1966), 73-88
  
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+
  
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<h5>재미있는 사실</h5>
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==관련된 항목들==
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<h5>관련된 단원</h5>
 
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<h5>관련된 고교수학 또는 대학수학</h5>
 
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<h5>관련된 다른 주제들</h5>
 
  
 
* [[수체의 class number]]
 
* [[수체의 class number]]
 
* [[정수계수 이변수 이차형식(binary integral quadratic forms)]]
 
* [[정수계수 이변수 이차형식(binary integral quadratic forms)]]
 +
* [[라마누잔의 class invariants]]
  
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+
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==매스매티카 파일 및 계산 리소스==
 +
* https://drive.google.com/file/d/0B8XXo8Tve1cxQVJtV1V4enN0Rmc/view
 +
  
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==수학용어번역==
  
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<h5>관련도서 및 추천도서</h5>
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* http://www.google.com/dictionary?langpair=en|ko&q=
 +
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]
 +
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 +
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
  
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* 도서내검색<br>
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
  
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<h5>참고할만한 자료</h5>
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==사전 형태의 자료==
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/
 
* http://en.wikipedia.org/wiki/Idoneal_number
 
* http://en.wikipedia.org/wiki/Idoneal_number
 +
 
* http://mathworld.wolfram.com/IdonealNumber.html
 
* http://mathworld.wolfram.com/IdonealNumber.html
  
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* http://www.wolframalpha.com/input/?i=
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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** http://www.research.att.com/~njas/sequences/?q=
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 +
  
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<h5>관련논문</h5>
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==관련논문==
  
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* [http://www.springerlink.com/content/04884m42j4760031/ Leonhard euler’s convenient number]<br>
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* [http://www.springerlink.com/content/04884m42j4760031/ Leonhard euler’s convenient number]
 
** [[3259130/attachments/2438477|Euler_s_convenient_numbers.pdf]]
 
** [[3259130/attachments/2438477|Euler_s_convenient_numbers.pdf]]
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** Günther Frei, The Mathematical Intelligencer, Volume 7, Number 3 / 1985년 9월
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** Günther Frei, The Mathematical Intelligencer, Volume 7, Number 3 / 1985년 9월
−
* Chowla, S. and Briggs, W. E.: On discriminants of binary quadratic forms with a single class in each genus. Canadian J. Math. 6 (1954), 463-470
+
* '''[Weinberger73]'''[http://matwbn.icm.edu.pl/tresc.php?wyd=6&tom=22 Exponents of the class groups of complex quadratic fields]
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* Grosswald, E.: Negative discriminants of binary quadratic forms with one class in each genus. Acta Arithmetica 8 (1963), 295-306
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** Weinberger, P. J., Acta Arithmetica 22 (1973), 117-124
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* Weinberger, P. J.: Exponents of the class groups of complex quadratic fields. Acta Arithmetica 22 (1973), 117-124
+
* [http://www.jstor.org/stable/2037547 On the exponent of the ideal class groups of complex quadratic fields]
−
 
+
** D. W. Boyd and H. Kisilevsky, Proc. Amer. Math. Soc. 31 (1972), 433-436.
−
 
+
* Negative discriminants of binary quadratic forms with one class in each genus
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+
** Grosswald, E, Acta Arithmetica 8 (1963), 295-306
−
 
+
* '''[Chowla1954]'''On discriminants of binary quadratic forms with a single class in each genus
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** Chowla, S. and Briggs, W. E. Canadian J. Math. 6 (1954), 463-470
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<h5>블로그</h5>
 
  
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* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=idoneal+number
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q3879415 Q3879415]
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===Spacy 패턴 목록===
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* [{'LOWER': 'idoneal'}, {'LEMMA': 'number'}]

2021년 2월 17일 (수) 05:54 기준 최신판

개요

  • 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
  • [Chowla1954] 는 이 목록이 유한임을 증명
  • 이 65개의 목록이 완전한 목록인지는 아직 미해결.
  • 복소이차수체의 class group의 exponent가 2 이하인 경우를 통해 이해할 수 있음
  • 이러한 성질을 갖는 복소이차수체는 지금까지 알려진 경우 외에 많아야 하나가 더 있을 수 있음이 증명, 디리클레 L-함수에 대한 일반화된 리만가설이 성립할 경우, 문제 해결 [Weinberger73]



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

  • 큰 소수를 찾아내는데 활용\[18518809=197^2+1848\cdot 100^2\]
  • 1848이 convenient 임을 이용하여 18518809라는 당시로서는 큰 소수를 찾아냄



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 = 18, 45, 72
  • If m > 1 is convenient and \(m \equiv 1 \pmod 4\) , then 4m is not convenient.
    • m = 5, 9, 13, 21, 25, 33, 37, 45, 57, 85, 93, 105, 133, 165, 177, 253, 273, 345, 357, 385, 1365
    • 4m = 20, 36, 52, 84, 100, 132, 148, 180, 228, 340, 372, 420, 532, 660, 708, 1012, 1092, 1380, 1428, 1540, 5460
  • If m is convenient and \(m \equiv 2 \pmod 4\), then 4m is convenient.
    • m = 2, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462
    • 4m = 8, 24, 40, 72, 88, 120, 168, 232, 280, 312, 408, 520, 760, 840, 1320, 1848
  • If m is convenient and \(m \equiv 8 \pmod {16}\), then 4m is not convenient.
    • m = 8, 24, 40, 72, 88, 120, 168, 232, 280, 312, 408, 520, 760, 840, 1320, 1848
    • 4m = 32, 96, 160, 288, 352, 480, 672, 928, 1120, 1248, 1632, 2080, 3040, 3360, 5280, 7392
  • If m is convenient and \(m \equiv 16 \pmod {32}\), then 4m is not convenient.
    • m = 16, 48, 112, 240
    • 4m = 64, 192, 448, 960
  • If m is convenient and \(m + x^2 = p^2 < 4m\) 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 discriminant \(\Delta = -4m\) 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 discriminant \(\Delta = -4m\) 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 1 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 number of such convenient numbers
1 1,2,3,4,7 5
2 5,6,8,9,10,12,13,15,16,18,22,25,28,37,58 15
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 24
8 105,120,165,168,210,240,273,280,312,330,345,357,385,408,462,520,760 17
16 840,1320,1365,1848 4



convenient number에 대한 이차형식의 목록

n=1,{x^2+y^2} n=2,{x^2+2 y^2} n=3,{x^2+3 y^2} n=4,{x^2+4 y^2} n=5,{x^2+5 y^2,2 x^2+2 x y+3 y^2} n=6,{x^2+6 y^2,2 x^2+3 y^2} n=7,{x^2+7 y^2} n=8,{x^2+8 y^2,3 x^2+2 x y+3 y^2} n=9,{x^2+9 y^2,2 x^2+2 x y+5 y^2} n=10,{x^2+10 y^2,2 x^2+5 y^2} n=12,{x^2+12 y^2,3 x^2+4 y^2} n=13,{x^2+13 y^2,2 x^2+2 x y+7 y^2} n=15,{x^2+15 y^2,3 x^2+5 y^2} n=16,{x^2+16 y^2,4 x^2+4 x y+5 y^2} n=18,{x^2+18 y^2,2 x^2+9 y^2} n=21,{x^2+21 y^2,2 x^2+2 x y+11 y^2,3 x^2+7 y^2,5 x^2+4 x y+5 y^2} n=22,{x^2+22 y^2,2 x^2+11 y^2} n=24,{x^2+24 y^2,3 x^2+8 y^2,4 x^2+4 x y+7 y^2,5 x^2+2 x y+5 y^2} n=25,{x^2+25 y^2,2 x^2+2 x y+13 y^2} n=28,{x^2+28 y^2,4 x^2+7 y^2} n=30,{x^2+30 y^2,2 x^2+15 y^2,3 x^2+10 y^2,5 x^2+6 y^2} n=33,{x^2+33 y^2,2 x^2+2 x y+17 y^2,3 x^2+11 y^2,6 x^2+6 x y+7 y^2} n=37,{x^2+37 y^2,2 x^2+2 x y+19 y^2} n=40,{x^2+40 y^2,4 x^2+4 x y+11 y^2,5 x^2+8 y^2,7 x^2+6 x y+7 y^2} n=42,{x^2+42 y^2,2 x^2+21 y^2,3 x^2+14 y^2,6 x^2+7 y^2} n=45,{x^2+45 y^2,2 x^2+2 x y+23 y^2,5 x^2+9 y^2,7 x^2+4 x y+7 y^2} n=48,{x^2+48 y^2,3 x^2+16 y^2,4 x^2+4 x y+13 y^2,7 x^2+2 x y+7 y^2} n=57,{x^2+57 y^2,2 x^2+2 x y+29 y^2,3 x^2+19 y^2,6 x^2+6 x y+11 y^2} n=58,{x^2+58 y^2,2 x^2+29 y^2} n=60,{x^2+60 y^2,3 x^2+20 y^2,4 x^2+15 y^2,5 x^2+12 y^2} n=70,{x^2+70 y^2,2 x^2+35 y^2,5 x^2+14 y^2,7 x^2+10 y^2} n=72,{x^2+72 y^2,4 x^2+4 x y+19 y^2,8 x^2+9 y^2,8 x^2+8 x y+11 y^2} n=78,{x^2+78 y^2,2 x^2+39 y^2,3 x^2+26 y^2,6 x^2+13 y^2} n=85,{x^2+85 y^2,2 x^2+2 x y+43 y^2,5 x^2+17 y^2,10 x^2+10 x y+11 y^2} n=88,{x^2+88 y^2,4 x^2+4 x y+23 y^2,8 x^2+11 y^2,8 x^2+8 x y+13 y^2} n=93,{x^2+93 y^2,2 x^2+2 x y+47 y^2,3 x^2+31 y^2,6 x^2+6 x y+17 y^2} n=102,{x^2+102 y^2,2 x^2+51 y^2,3 x^2+34 y^2,6 x^2+17 y^2} n=105,{x^2+105 y^2,2 x^2+2 x y+53 y^2,3 x^2+35 y^2,5 x^2+21 y^2,6 x^2+6 x y+19 y^2,7 x^2+15 y^2,10 x^2+10 x y+13 y^2,11 x^2+8 x y+11 y^2} n=112,{x^2+112 y^2,4 x^2+4 x y+29 y^2,7 x^2+16 y^2,11 x^2+6 x y+11 y^2} n=120,{x^2+120 y^2,3 x^2+40 y^2,4 x^2+4 x y+31 y^2,5 x^2+24 y^2,8 x^2+15 y^2,8 x^2+8 x y+17 y^2,11 x^2+2 x y+11 y^2,12 x^2+12 x y+13 y^2} n=130,{x^2+130 y^2,2 x^2+65 y^2,5 x^2+26 y^2,10 x^2+13 y^2} n=133,{x^2+133 y^2,2 x^2+2 x y+67 y^2,7 x^2+19 y^2,13 x^2+12 x y+13 y^2} n=165,{x^2+165 y^2,2 x^2+2 x y+83 y^2,3 x^2+55 y^2,5 x^2+33 y^2,6 x^2+6 x y+29 y^2,10 x^2+10 x y+19 y^2,11 x^2+15 y^2,13 x^2+4 x y+13 y^2} n=168,{x^2+168 y^2,3 x^2+56 y^2,4 x^2+4 x y+43 y^2,7 x^2+24 y^2,8 x^2+21 y^2,8 x^2+8 x y+23 y^2,12 x^2+12 x y+17 y^2,13 x^2+2 x y+13 y^2} n=177,{x^2+177 y^2,2 x^2+2 x y+89 y^2,3 x^2+59 y^2,6 x^2+6 x y+31 y^2} n=190,{x^2+190 y^2,2 x^2+95 y^2,5 x^2+38 y^2,10 x^2+19 y^2} n=210,{x^2+210 y^2,2 x^2+105 y^2,3 x^2+70 y^2,5 x^2+42 y^2,6 x^2+35 y^2,7 x^2+30 y^2,10 x^2+21 y^2,14 x^2+15 y^2} n=232,{x^2+232 y^2,4 x^2+4 x y+59 y^2,8 x^2+29 y^2,8 x^2+8 x y+31 y^2} n=240,{x^2+240 y^2,3 x^2+80 y^2,4 x^2+4 x y+61 y^2,5 x^2+48 y^2,12 x^2+12 x y+23 y^2,15 x^2+16 y^2,16 x^2+16 x y+19 y^2,17 x^2+14 x y+17 y^2} n=253,{x^2+253 y^2,2 x^2+2 x y+127 y^2,11 x^2+23 y^2,17 x^2+12 x y+17 y^2} n=273,{x^2+273 y^2,2 x^2+2 x y+137 y^2,3 x^2+91 y^2,6 x^2+6 x y+47 y^2,7 x^2+39 y^2,13 x^2+21 y^2,14 x^2+14 x y+23 y^2,17 x^2+8 x y+17 y^2} n=280,{x^2+280 y^2,4 x^2+4 x y+71 y^2,5 x^2+56 y^2,7 x^2+40 y^2,8 x^2+35 y^2,8 x^2+8 x y+37 y^2,17 x^2+6 x y+17 y^2,19 x^2+18 x y+19 y^2} n=312,{x^2+312 y^2,3 x^2+104 y^2,4 x^2+4 x y+79 y^2,8 x^2+39 y^2,8 x^2+8 x y+41 y^2,12 x^2+12 x y+29 y^2,13 x^2+24 y^2,19 x^2+14 x y+19 y^2} n=330,{x^2+330 y^2,2 x^2+165 y^2,3 x^2+110 y^2,5 x^2+66 y^2,6 x^2+55 y^2,10 x^2+33 y^2,11 x^2+30 y^2,15 x^2+22 y^2} n=345,{x^2+345 y^2,2 x^2+2 x y+173 y^2,3 x^2+115 y^2,5 x^2+69 y^2,6 x^2+6 x y+59 y^2,10 x^2+10 x y+37 y^2,15 x^2+23 y^2,19 x^2+8 x y+19 y^2} n=357,{x^2+357 y^2,2 x^2+2 x y+179 y^2,3 x^2+119 y^2,6 x^2+6 x y+61 y^2,7 x^2+51 y^2,14 x^2+14 x y+29 y^2,17 x^2+21 y^2,19 x^2+4 x y+19 y^2} n=385,{x^2+385 y^2,2 x^2+2 x y+193 y^2,5 x^2+77 y^2,7 x^2+55 y^2,10 x^2+10 x y+41 y^2,11 x^2+35 y^2,14 x^2+14 x y+31 y^2,22 x^2+22 x y+23 y^2} n=408,{x^2+408 y^2,3 x^2+136 y^2,4 x^2+4 x y+103 y^2,8 x^2+51 y^2,8 x^2+8 x y+53 y^2,12 x^2+12 x y+37 y^2,17 x^2+24 y^2,23 x^2+22 x y+23 y^2} n=462,{x^2+462 y^2,2 x^2+231 y^2,3 x^2+154 y^2,6 x^2+77 y^2,7 x^2+66 y^2,11 x^2+42 y^2,14 x^2+33 y^2,21 x^2+22 y^2} n=520,{x^2+520 y^2,4 x^2+4 x y+131 y^2,5 x^2+104 y^2,8 x^2+65 y^2,8 x^2+8 x y+67 y^2,13 x^2+40 y^2,20 x^2+20 x y+31 y^2,23 x^2+6 x y+23 y^2} n=760,{x^2+760 y^2,4 x^2+4 x y+191 y^2,5 x^2+152 y^2,8 x^2+95 y^2,8 x^2+8 x y+97 y^2,19 x^2+40 y^2,20 x^2+20 x y+43 y^2,29 x^2+18 x y+29 y^2} n=840,{x^2+840 y^2,3 x^2+280 y^2,4 x^2+4 x y+211 y^2,5 x^2+168 y^2,7 x^2+120 y^2,8 x^2+105 y^2,8 x^2+8 x y+107 y^2,12 x^2+12 x y+73 y^2,15 x^2+56 y^2,20 x^2+20 x y+47 y^2,21 x^2+40 y^2,24 x^2+35 y^2,24 x^2+24 x y+41 y^2,28 x^2+28 x y+37 y^2,29 x^2+2 x y+29 y^2,31 x^2+22 x y+31 y^2} n=1320,{x^2+1320 y^2,3 x^2+440 y^2,4 x^2+4 x y+331 y^2,5 x^2+264 y^2,8 x^2+165 y^2,8 x^2+8 x y+167 y^2,11 x^2+120 y^2,12 x^2+12 x y+113 y^2,15 x^2+88 y^2,20 x^2+20 x y+71 y^2,24 x^2+55 y^2,24 x^2+24 x y+61 y^2,33 x^2+40 y^2,37 x^2+14 x y+37 y^2,40 x^2+40 x y+43 y^2,41 x^2+38 x y+41 y^2} n=1365,{x^2+1365 y^2,2 x^2+2 x y+683 y^2,3 x^2+455 y^2,5 x^2+273 y^2,6 x^2+6 x y+229 y^2,7 x^2+195 y^2,10 x^2+10 x y+139 y^2,13 x^2+105 y^2,14 x^2+14 x y+101 y^2,15 x^2+91 y^2,21 x^2+65 y^2,26 x^2+26 x y+59 y^2,30 x^2+30 x y+53 y^2,35 x^2+39 y^2,37 x^2+4 x y+37 y^2,42 x^2+42 x y+43 y^2} n=1848,{x^2+1848 y^2,3 x^2+616 y^2,4 x^2+4 x y+463 y^2,7 x^2+264 y^2,8 x^2+231 y^2,8 x^2+8 x y+233 y^2,11 x^2+168 y^2,12 x^2+12 x y+157 y^2,21 x^2+88 y^2,24 x^2+77 y^2,24 x^2+24 x y+83 y^2,28 x^2+28 x y+73 y^2,33 x^2+56 y^2,43 x^2+2 x y+43 y^2,44 x^2+44 x y+53 y^2,47 x^2+38 x y+47 y^2}


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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  • [{'LOWER': 'idoneal'}, {'LEMMA': 'number'}]
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