×

# Greatest integer and factorial part equation

Calculation of Real $$(x,y,z)$$\$ in

$$x[x]+z\{z\}-y\{y\} = 0.16$$

$$y[y]+x\{x\}-z\{z\} = 0.25$$

$$z[z]+y\{y\}-x\{x\} = 0.49$$

where $$[x] =$$Greatest Integer of $$x$$ and $$\{x\} =$$ fractional part of $$x$$

Note by Jagdish Singh
3 years, 8 months ago

Sort by:

(POST#2) - In post #1, we have discovered that one and only one of $$[x]$$, $$[y]$$ and $$[z]$$ must be $$1$$ or $$-1$$.

• Case 1: $$[x]=1$$,$$[y]=0$$,$$[z]=0$$
• (1): $$x+z^2-y^2=0.16$$ ---(4)
• (2): $$x(x-1)-z^2=0.25$$ ---(5)
• (3): $$y^2-x(x-1)=0.49$$ ---(6)
• (4)+(5)+(6): $$x=0.9$$, which contradicts the prerequisite.

• Case 2: $$[x]=-1$$,$$[y]=0$$,$$[z]=0$$
• (1): $$-x+z^2-y^2=0.16$$ ---(4)
• (2): $$x(x+1)-z^2=0.25$$ ---(5)
• (3): $$y^2-x(x+1)=0.49$$ ---(6)
• (4)+(5)+(6): $$x=-0.9$$ ---(7)
• (7) into (6): $$y=\sqrt{0.4}$$
• (7) into (5): $$z=\sqrt{-0.34}$$, which is not real.

• Case 3: $$[x]=0$$,$$[y]=1$$,$$[z]=0$$
• (1): $$z^2-y(y-1)=0.16$$ ---(4)
• (2): $$y+x^2-z^2=0.25$$ ---(5)
• (3): $$y(y-1)-x^2)=0.49$$ ---(6)
• (4)+(5)+(6): $$y=0.9$$, which contradicts the prerequisite.
• In fact, I expect case 5 to fail as well, because it would yield $$z=0.9$$.
• Therefore, I will skip case 5.

• Case 4: $$[x]=0$$,$$[y]=-1$$,$$[z]=0$$
• (1): $$z^2-y(y+1)=0.16$$ ---(4)
• (2): $$-y+x^2-z^2=0.25$$ ---(5)
• (3): $$y(y+1)-x^2=0.49$$ ---(6)
• (4)+(5)+(6): $$y=-0.9$$ ---(7)
• (7) into (6): $$x=\pm\sqrt{-0.56}$$, which is not real.

• Case 5: $$[x]=0$$,$$[y]=0$$,$$[z]=1$$ (skipped)

• Case 6: $$[x]=0$$,$$[y]=0$$,$$[z]=-1$$
• (1): $$z(z+1)+y^2=0.16$$ ---(4)
• (2): $$x^2-z(z+1)=0.25$$ ---(5)
• (3): $$-z+y^2-x^2)=0.49$$ ---(6)
• (4)+(5)+(6): $$z=-0.9$$ ---(7)
• (7) into (5): $$x=\sqrt{0.34}$$
• (7) into (4): $$y=\sqrt{0.25}$$

• $$x=\sqrt{0.34}, y=0.5, z=-0.9$$.
· 2 years, 8 months ago

(POST#1)

• Let $$[x],\{x\},[y],\{y\},[z],\{z\}$$ be $$a,b,c,d,e,f$$ respectively.
• $$\left\{\begin{array}{lclr}a^2+ab+ef+f^2-cd-d^2&=&0.16&\mbox{---(1)}\\c^2+cd+ab+b^2-ef-f^2&=&0.25&\mbox{---(2)}\\e^2+ef+cd+d^2-ab-b^2&=&0.49&\mbox{---(3)}\end{array}\right.$$
• (1)+(2)+(3): $$a^2+ab+c^2+cd+e^2+ef = 0.9 \mbox{---(4)}$$
• (4)-(2): $$a^2-b^2+(e+f)^2=0.65 \mbox{---(5)}$$
• (4)-(3): $$c^2-d^2+(a+b)^2=0.41 \mbox{---(6)}$$
• (4)-(1): $$e^2-f^2+(c+d)^2=0.74 \mbox{---(7)}$$
• Since $$0\le b^2,d^2,f^2<1$$,
• (5): $$a^2+(e+f)^2=0.65+b^2<1.65$$ ---(8)
• (6): $$c^2+(a+b)^2=0.41+d^2<1.41$$ ---(9)
• (7): $$e^2+(c+d)^2=0.74+f^2<1.74$$ ---(10)
• Since $$0\le b,d,f<1$$,
• (8): $$a^2+e^2<1.65$$ ---(11)
• (9): $$c^2+a^2<1.41$$ ---(12)
• (10): $$e^2+c^2<1.74$$ ---(13)
• Since $$a$$, $$c$$, and $$e$$ are integers, they can only be $$0$$ or $$\pm1$$ according to (11), (12) and (13).
• Moreover, there can be at most one of $$a$$, $$c$$, or $$e$$ which is $$\pm1$$, or else (11), (12) or (13) will not be satisfied.
• However, of all three of them are zero, (4) will not be satisfied.
• In the next post, I shall explore all the three cases.
· 2 years, 8 months ago