Equation in real numbers // Ligning i reelle tal

 It's problem E : 17 361  in the picture.

 In translation : "E:17361.  Solve the equation  $[x]=\sqrt{\{x\}\cdot (x+1)}$  in the set of

                            real numbers.

 [Author]  Ionuţ ALEXUC Timişoara"

Remark CiP :  No matter how often they are used, the notations  $\{x\},\;[x]$  no longer have any presentation.  : {x} is fractional part and [x] is the integer part of the real number x.

In my answer S means the set of solutions to the given equation. The problem is proposed for 7th grade (students aged 12-14).

ANSWER CiP

$S=\{0,\;\sqrt{2}\}$

               

                               Solution CiP

               The domain on which the equation is defined is the interval  $x\in [-1,\;+\infty)$. (Because  $\{x\}\geqslant 0$)

          But if  $x<0$ , then  $[x]\leqslant -1$,  so strictly negative and then the equation cannot have such solutions. It remains that  $x\geqslant 0$.

           For  $0\leqslant x<1$  we have  $[x]=0$, and how  $x+1\neq 0$ it must  $\{x\}=0$, so  $x=0$.

           For  $x\geqslant 1$ , from  $x-1<[x]\leqslant x\;\Rightarrow\;\;x-1<\sqrt{\{x\}\cdot (x+1)}\leqslant x$

Hence  $(x-1)^2<\{x\}\cdot (x+1)\;\;\Rightarrow\;\{x\}>\frac{(x-1)^2}{x+1}$. But  $\{x\}<1$  so  $1>\frac{(x-1)^2}{x+1}$

which is equivalent to  $x+1>x^2-2x+1\;\Leftrightarrow\;3x>x^2\;\Leftrightarrow\;3>x$.

          So we can have  $[x]=1\;or\;[x]=2$.

          If  $[x]=1$  the equation is written  $1=\sqrt{\{x\}\cdot (1+\{x\}+1)}$

$\Leftrightarrow\;1=\{x\}\cdot(\{x\}+2)\;\Leftrightarrow\;\{x\}^2+2\{x\}=1\;\Leftrightarrow\;(\{x\}+1)^2=2\;\Leftrightarrow\;\{x\}+1=\sqrt{2}.$

Hence  $x=[x]+\{x\}=1+\{x\}=\sqrt{2}$.

          If  $[x]=2$  the equation is written  $2=\sqrt{\{x\}\cdot (2+\{x\}+1)}$

$\Leftrightarrow\;4=\{x\}\cdot (\{x\}+3)\;\Leftrightarrow\;\{x\}^2+3\{x\}=4\;\Rightarrow\;\{x\}=1$  what is not possible.

Both values ​​found for  $x$  verify the equation.

$\blacksquare$

Reading GROSSWALD Emil "Representations of Integers as Sums of Squares" // GROSSWALD Emil "Representations of Integers as Sums of Squares" lesen

 Ein sehr schwer zu findendes Buch. Du kannst es dir hier ansehen.

A very hard-to-find book. You can view it here.

I have some of the books mentioned below in my Electronic Library.

The first encounter with the problem happened while reading SIERPIŃSKI W. Elementary Theory of Numbers (1988). On page 449 we have Problem 2(stated as Exercise) :

   " 2.   Find the complex integers  $x+y\imath$ which

 are representable as sums of the squares of two

 complex integers"

The exercise only has the answer, without any indication, not even bibliographical, of how to obtain it.

Read this solution yourself !!

A necessary condition is:  $y$  is an even number.

Then, for the number  $x+2y\imath$  we have :

$x+2y\imath\;=$  sums of the squares of two complex integers  $\Leftrightarrow$

$\Leftrightarrow$   NOT  both of  $\frac{x}{2}\;and\;y$  are odd integers

This formulation of the necessary and sufficient condition is inspired by NIVEN's article. The discussion is presented in more detail in GROSSWALD, pages 194-195.

        SIERPIŃSKI formulates, as I invited you to see, the necessary and sufficient condition in this way :

in  $x+y\imath\;\;y$  should be even and, in the case  $x=4t+2,\;\;y$  should be divisible by 4

Moreover, the corresponding writing is also presented, which no matter how much I searched, I could not find the way in which it was discovered.

ANSWER WS

$4t+4u\imath=[(t+1)+u\imath]^2+[u+(1-t)\imath]^2 \tag{1}$

$4t+(4u+2)\imath=[(t+u+1)+(u+1-t)\imath]^2+[(t-u)+(t+u)\imath]^2 \tag{2}$

$(2t+1)+2u\imath=[(t+1)+u\imath]^2+[u-t\imath]^2 \tag{3}$

$(4t+2)+4u\imath=[(t+u+1)+(u-t)\imath]^2+[(t-u+1)+(t+u)\imath]^2 \tag{4}$

                    Examples CiP

          Formulas (1)-(4) cover all possible cases of the numbers  $x+2y\imath$.  Indeed:

     -  when  $x \neq 4t+2$  we can have

                                                 $x=4t$,  when for  $y=2u$  we have (1)

 and for  $y=2u+1$  we have (2)                 

                                                  $x=4t+1\;\;or\;\;x=4t+3$, written together as

                                                            $x=2t+1$, and  $y$  can be any, having (3)

     -  when  $x=4t+2$  then necessarily  $y=2u$, and we have (4)

                 1.       $6\imath=(2+2\imath)^2+(-1+\imath)^2$         $t=0,\;u=1\;\;in\;(2)$;

                 2.    $5-6\imath=(3-3\imath)^2+(-3-2\imath)^2$      $t=2,\;u=-3\;\;in\;(3)$;

                 3.  $-2+4\imath=(1+2\imath)^2+(-1)^2$          $t=-1,\;u=1\;\;in\;(4)$;

                 4.    $3-4\imath=(2-2\imath)^2+(-2-\imath)^2$     $t=1,\;u=-2\;\;in\;(3)$;

                 5.  $-4+8\imath=(2\imath)^2+(2-2\imath)^2$            $t=-1,\;u=2\;\;in\;(1)$.

               REMARK CiP

                  Regarding the number of writings as the sum of two squares, we make the following observations :

            i)  One would expect that Gaussian integers, which are prime (see here), would essentially have a unique writing as the sum of two squares.  

           ii)  If a number is not prime, then it could (not necessarily) have more than one expression as a sum of squares. It all depends on the number of prime divisors, of a certain type, of that number.

             For example, let the Gaussian primes be

$1+2\imath=(1+\imath)^2+1^2$

$3-2\imath=(2-\imath)^2+(-1-\imath)^2$

     (I applied (3) to write them as a sum of squares.) Their product is

$(1+2\imath)\cdot (3-2\imath)=7+4\imath$

and has, according to  (3)  with $t=3,\;u=2$, the representation

$7+4\imath=(4+2\imath)^2+(2-3\imath)^2 \tag{e}$

On the other hand we have

$7+4\imath=[(1+\imath)^2+1^2]\cdot [(2-\imath)^2+(-1-\imath)^2]$

and if, above, we apply one or the other of the formulas

$(a^2+b^2)\cdot (c^2+d^2)=(ac+bd)^2+(ad-bc)^2=(ad+bc)^2+(ac-bd)^2$

 we have

$(I)\;\;7+4\imath=[(1+\imath)(2-\imath)+1(-1-\imath)]^2+[(1+\imath)(-1-\imath)-1(2-\imath)]^2=$

$=(2+1+2\imath-\imath-1-\imath)^2+(-1+1-\imath-\imath-2+\imath)^2=2^2+(-2-\imath)^2$

or

$(II)\;\;7+4\imath=[(1+\imath)(-1-\imath)+1(2-\imath)]^2+[(1+\imath)(2-\imath)-1(-1-\imath)]^2=$

$=(-2\imath+2-\imath)^2+(2+2\imath-\imath+1+1+\imath)^2=(2-3\imath)^2+(4+2\imath)^2$

which is exactly  (e). Thus we have the double representation

$7+4\imath=2^2+(2+\imath)^2=(2-3\imath)^2+(4+2\imath)^2$

Squares in the Set of Gauss Integers and the Pythagorean Triple Equation // กำลังสองในเซตของจำนวนเต็มเกาส์และสมการสามเท่าของพีทาโกรัส

 

          Complex numbers of the form  $a+\imath b$  with  $a,b \in \mathbb{Z}$  are called Gaussian integers.

          Here we will consider the problem of finding Gaussian integers that are "perfect squares". For example, we are interested in when the result  $\sqrt{a+\imath b}$ is still a Gaussian integer.

           The problem is solved for example in W. Sierpinski (chapter XIII is dedicated to Gauss integers), page 450, Problem 4. We see that this problem is related to Pythagorean triples. This problem is not completed, unlike Problem 2 about representing as sums of two squares.

          What is said here is only a necessary and sufficient condition that solves our problem :

      "The complex integer  $a+\imath b$  is the square of a complex integer if and only if 

$a^2+b^2=c^2,\;\;\;c+a=2x^2,\;\;\;c-a=2y^2$

   where  $c$ is a natural number and  $x,\;y$  are rational integers. Then

$a+\imath b=(\pm x\pm y\imath)^2$

   the sign should be identical if  $b>0$  and opposite if  $b<0$."

       So the perfect squares in the set of Gauss integers are related to the solutions of the equation of Pythagorean numbers. As shown in another post, for the numbers  $a$  and  $b$  we have the possibilities :

$a=d\cdot 2st,\;\;b=d\cdot (s^2-t^2) \tag{1}$

$a=d\cdot (s^2-t^2),\;\;b=d\cdot 2st \tag{2}$

where  $d\in \mathbb{Z}$  and  $s,\;t$  are coprime integers of opposite parity.

          In the first case  $c+a=d\cdot (s^2+t^2)+d\cdot 2st=d\cdot (s+t)^2$, and $c-a=d\cdot (s^2+t^2)-d\cdot 2st=d\cdot (s-t)^2$. Hence

$\frac{c\pm a}{2}=\frac{d}{2}\cdot (s \pm t)^2 \tag{3}$

          In the second case  $c+a=d\cdot (s^2+t^2)+d\cdot (s^2-t^2)=2ds^2$, and  $c-a=d\cdot (s^2+t^2)-d\cdot (s^2-t^2)=2dt^2$,  hence

$\frac{c+a}{2}=d\cdot s^2,\;\; \frac{c-a}{2}=d\cdot t^2 \tag{4}$

So for  $\frac{c\pm a}{2}$  to be perfect squares,  $d$  must be  $2\alpha^2$ in (3) and  $\alpha^2$ in (4).

Answer CiP: The perfect squares in the set of Gaussian integers are

$4\alpha^2st+2\alpha^2(s^2-t^2)\imath=\left ( \pm\alpha[(s+t)+(s-t)\imath]\right )^2$

$\alpha^2(s^2-t^2)+2\alpha^2st \imath=\left (\pm \alpha [s+t\imath] \right )^2$

where  $\alpha \in \mathbb{Z}$, and  $s,t$ are coprime integers of opposite parity. 

          Examples CiP

$3+4\imath=(2+\imath)^2=(-2-\imath )^2$ :   $3^2+4^2=5^2,\;\frac{5\pm 3}{2}\in \{4,\;1\}$

$2\imath=(1+\imath)^2=(-1-\imath)^2$  :   $0^2+2^2=2^2,\;\frac{2\pm 0}{2}=1$

$5\pm 12 \imath=(3\pm 2\imath)^2$   :   $5^2+12^2=13^2,\;\frac{13\pm 5}{2}\in \{9,\;4\}$

$12\pm 5\imath \neq perfect\;squares$   :    $12^2+5^2=13^2$  although  $\frac{13\pm 12}{2}\in \left \{\frac{25}{2},\;\frac{1}{2}\right \}$

$24\pm 10\imath=(5\pm \imath)^2$   :   $24^2+10^2=26^2,\;\frac{26\pm 24}{2}\in\{25,\;1\}$

Problem E : 17346

 From GMB 11/2025, page 554. In translation :

  "E:17346.  A natural number  $n$ gives when divided by 7 the remainder 4

                       and when divided by 9 the remainder 5. Find out what remainder is

                       obtained  when dividing  $n$  by 63.

 [Author] Ștefan GOBLEJ, Curtea de Argeș"

ANSWER CiP

32

                            Solution CiP

                    Let  $a$ and  $b$  be the quotients of the divisions of  $n$  by 7 and 9 respectively. Then

$n=7\cdot a+4,\;\;\;n=9\cdot b+5 \;\;\;\;\;a,\;b\in \mathbb{N}\tag{1}$

The equation  $7\cdot a+4=9\cdot b+5$   is equivalent to

$7\cdot a-9\cdot b=1\;\;\;\;\;\;a,b \in \mathbb{N} \tag{2}$

Noting that  $7\cdot 4-9\cdot 3=28-27=1$  , equation  (2)  has a particular solution  $a_0=4,\;b_0=3$.

So the general solution of this [linear Diophantine] equation is

$a=4+9\cdot k,\;\;b=3+7\cdot k,\;\;\;k \in \mathbb{N} \tag{3}$

Substituting the number  $a$  from (3) into (1) we obtain  $n=7\cdot (4+9\cdot k)+4$  so  

$$n=63 \cdot k+32$$

and from this it follows that the remainder of dividing the number  $n$  by 63 is 32.

$\blacksquare$

             Remark CiP  We could have thought like that too, without the theory of Diophantine equations :

$7\cdot a\overset{(2)}{=}1+9\cdot b=1+2\cdot b+7\cdot b\;\Rightarrow\; 7\mid 1+2\cdot b\;\Rightarrow\; 2\cdot b+1=7\cdot k_1,\;k_1 \in \mathbb{N}$. But in the equality  $2\cdot b=7\cdot k_1-1$  we must have  $k_1=odd\;number,\;\; k_1=2\cdot k+1$  so  $2\cdot b=7(2\cdot k+1)-1=14 \cdot k+6$  and from here we obtain  $b=7\cdot k+3$  that is precisely  (3).

          COPILOT's solution : 

We write the numbers that give remainder of 4 when divided by 7 :

4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, ...

We check which of them gives a remainder of 5 when divided by 9 :

4  mod 9  = 4

11  mod 9  = 2

18  mod 9  = 0

25  mod 9  =7

32  mod  9  =5  :  BINGO

(CiP note : After at most nine (=9) attempts, the result is definitely found, whether it exists or not.)

So the number that satisfies both conditions is  $n \equiv 32\;\;(mod\;63)$  , hence the remainder when dividing  $n$  by 63 is 32.

LUDO // oder Mensch ärgere dich nich // Another calculation problem with Vectors

           The game known in Romania as "Don't Be Angry, Brother!" it's basically LUDO. There is also a German version, mentioned in the title. I would have liked to say "Don't Be Upset, Sister!", because alongside Problem S:L25.296 in Post here, there is also its sister, Problem S:L25.295, which I initially neglected.

In translation :

"Let the triangle  $ABC$  and the points  $M\in (AB),\; N\in (AC), \;P\in BC$

  be such that  $\overrightarrow{BC}=2\overrightarrow{CP},\;\frac{MA}{MB}=\frac{1}{2},\;\;and\;\;\frac{NC}{AC}=\frac{2}{5}.$

     a) Express the vectors  $\overrightarrow{AM},\;\overrightarrow{AN}$  and  $\overrightarrow{AP}$ in terms of the vectors  $\overrightarrow{AB}\;and\;\overrightarrow{AC}$.

     b) Express the vectors  $\overrightarrow{MN}$  and  $\overrightarrow{MP}$  in terms of the vectors  $\overrightarrow{AB}\;and\;\overrightarrow{AC}$.

 c) Show that the points  $M,\;N,\;P$  are collinear and determine the ratio  $\frac{MN}{MP}.$

No author"

ANSWER CiP

a)  $\overrightarrow{AM}=\frac{1}{3}\overrightarrow{AB}$

$\overrightarrow{AN}=\frac{3}{5}\overrightarrow{AC}$

$\overrightarrow{AP}=-\frac{1}{2}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC}$

b)  $\overrightarrow{MN}=-\frac{1}{3}\overrightarrow{AB}+\frac{3}{5}\overrightarrow{AC}$

$\overrightarrow{MP}=-\frac{5}{6}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC}$

c)  $\overrightarrow{MN}=\frac{2}{5}\overrightarrow{MP}$

$\frac{MN}{MP}=\frac{2}{5}$

                         Solution CiP

          a) From  $\frac{MA}{MB}=\frac{1}{2}$  and  $M\in (AB)$  it follow  $\frac{\overline{MA}}{\overline{MB}}=\frac{-1}{2}$ and so  

$\frac{\overline{AM}}{\overline{AB}}=-\frac{\overline{MA}}{\overline{MB}-\overline{MA}}=-\frac{-1}{2-(-1)}=\frac{1}{3}$, hence  $\overrightarrow{AM}=\frac{1}{3}\overrightarrow{AB}.$

From  $\frac{NC}{AC}=\frac{2}{5}$  and  $N\in (AC)$  it follow  $\frac{\overline{NC}}{\overline{AC}}=\frac{2}{5}$  so

  $\frac{\overline{AN}}{\overline{AC}}=\frac{\overline{AC}+\overline{CN}}{\overline{AC}}=\frac{\overline{AC}-\overline{NC}}{\overline{AC}}=\frac{5-2}{5}=\frac{3}{5}$,  hence  $\overrightarrow{AN}=\frac{3}{5}\overrightarrow{AC}$

Finally  $\overrightarrow{AP}=\overrightarrow{AC}+\overrightarrow{CP}=\overrightarrow{AC}+\frac{1}{2}\overrightarrow{BC}=\overrightarrow{AC}+\frac{1}{2}(\overrightarrow{AC}-\overrightarrow{AB})=-\frac{1}{2}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC}$.

The same answer can be obtained by observing that  $\frac{\overline{PB}}{\overline{PC}}=\frac{3}{1}$  and we apply the formulas (1.1) and (3.2)

  $\overrightarrow{AP}=\frac{1}{1-3}\overrightarrow{AB}-\frac{3}{1-3}\overrightarrow{AC}.$

          b)  $\overrightarrow{MN}=\overrightarrow{AN}-\overrightarrow{AM}=\frac{3}{5}\overrightarrow{AC}-\frac{1}{3}\overrightarrow{AB}$  hence the answer.

     $\overrightarrow{MP}=\overrightarrow{AP}-\overrightarrow{AM}=\left (-\frac{1}{2}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC} \right )-\frac{1}{3}\overrightarrow{AB}=-\frac{5}{6}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC}$

          c)  With the results from  b)  we have

$\overrightarrow{MN}=-\frac{1}{3}\overrightarrow{AB}+\frac{3}{5}\overrightarrow{AC}=\frac{2}{5} \cdot \left (-\frac{5}{6}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC} \right )=\frac{2}{5}\overrightarrow{MP}$

hence the points  $M,\;N,\;P$  are collinear and  $\frac{\overline{MN}}{\overline{MP}}=\frac{2}{5}.$

     The collinearity of points  $M,\;N,\;P$  also results from observing the  $\overrightarrow{AP}=-\frac{1}{2}\overrightarrow{AB}+\frac{3}{2}\overrightarrow{AC}=-\frac{1}{2}\cdot 3\overrightarrow{AM}+\frac{3}{2}\cdot \frac{5}{3}\overrightarrow{AN}$  or,

$$\overrightarrow{AP}=\frac{5}{2}\overrightarrow{AN}-\frac{3}{2}\overrightarrow{AM}=\frac{1}{\frac{2}{5}}\overrightarrow{AN}+\left ( 1-\frac{1}{\frac{2}{5}}\right )\overrightarrow{AM}$$

to which we apply formulas  (1.1)  and  (2.2).

$\blacksquare$