## Thursday, January 12, 2012

### 724

724 = 22 x 181.

724 has a representation as a sum of two squares: 724 = 182 + 202.

724 divides 194 - 1.

724 is the number of perfect connected undirected simple graphs on seven nodes.

724 is the number of different arrangements of 10 non-attacking queens on a 10 x 10 chessboard.

724 Hapag is an asteroid discovered in 1911, lost, then rediscovered in 1988.

#### 1 comment:

Anonymous said...

Algunas propiedades analíticas del número 724
Orden multiplicativo:
Como 27^2>724>26^2, y los primos equidistantes son 29 y 23
29^15(mód.724) = 1 y 23^180(mód.724 )= 1 son ordenes multiplicativos.
Factorización gaussiana:
4 = [(1+i)(1-i)]^2 = [(1+i)(1+i)(-i)]^2
181 = (10+9i)(10-9i) = (10+9i)(9+10i)(-i)
724 = [(1+i)(1-i)]^2[(10+9i)(10-9i)]
724 = [(1+i)(1+i)(-i)]^2[(10+9i)(9+10i)(-i)]
http://hojamat.es/parra/gaussiana.pdf
724 = x^2+Dy^2 = 11^2+67*3^2 = 19^2+3*11^2
724 = (11+3(-67)^(1/2)) = (11-3(-67)^(1/2)) = 11+-(-67)^(1/2)
724 = (19+11(-3)^(1/2)) = (19-11(-3)^(1/2)) = 19+-11(-67)^(1/2)
724 = x^2+Dy^2 = 29^2-13*3^2 = 57^2-101*5^2
724 = (29+3(13)^(1/2))(29-3(13)^(1/2)) = 29+-3(13)^(1/2)
724 = (57+5(101)^(1/2))(57-5(101)^(1/2)) = 57+-5(101)^(1/2)
Grupos multiplicativos:
Sea z = 724+899t y x = 28+29t e y = 11+31t los elementos aditivos
de dicho grupo, donde
mcd (29,31) = 1 = 29(15)+31(-14), entonces:
x = 28+29t = 29(15)(11+31t) = 290(mód. 899)
x = 11+31t = 31(-14)(28+29t) = 434(mód. 899)
z = 724+899t = 290+434 = 724(mód. 899)
f(724) = f(290) + f(434)
Sistemas multivariables modulares:
724 (mód. 23) = 5x^2+12y =11 (mód. 23). Para (*)
x = 6+23t = 5(6+23t)^2 = 19 (mód. 23)
y = 7+23t = 12(7+23t) = 15 (mód. 23)
z = 724+23t = 19+15 = 11 (mód. 23)
(*) Las soluciones están comprendidas entre {1,2,...,23-1} que es el sistema
completo de restos respecto al módulo 23.
http://hojamat.es/parra/iniparra.htm
Rafael Parra Machío