오일러의 convenient number ( Idoneal number)

수학노트
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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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