How-many-ways-can-2018-be-expressed-as-the-sum-of-two-squares-

Question Number 110895 by Aina Samuel Temidayo last updated on 31/Aug/20

How many ways can 2018 be

expressed as the sum of two squares ?

Answered by ajfour last updated on 31/Aug/20

43^{2} + 13^{2} = 2018

Answered by floor(10²Eta[1]) last updated on 31/Aug/20

y = \sqrt{2018 - x^{2}} \Rightarrow x < 45 and y < 45

x^{2} + y^{2} = 2018 \Rightarrow {(x + y)}^{2} = 2 (1009 + xy)

{(x + y)}^{2} is even \Rightarrow (x + y) is even .

suppose x and y are even

x = 2 a and y = 2 b

{(2 a + 2 b)}^{2} = 2 (1009 + (2 a) (2 b))

\Rightarrow 2 (a^{2} + b^{2}) = 1009

but 1009 is not even so x and y {can}^{'} t be even .

\Rightarrow x and y are odd .

\gcd (x, y) = d \Rightarrow x = ds, y = dt, \gcd (s, t) = 1

d^{2} (s^{2} + t^{2}) = 2018

d^{2} ∣ 2018 \Rightarrow d^{2} \in {1, 2, 1009, 2018} \Rightarrow d = 1

so x and y are coprimes

x, y \in (1, 3, 5, 7, \dots, 43)

with that you can try to solve the problem

by testing but i will continue to think

and try to get a better way

Commented by Aina Samuel Temidayo last updated on 31/Aug/20

Thanks

Answered by Rasheed.Sindhi last updated on 01/Sep/20

(i) unit - digits .

(ii) squares of unit - digits .

(iii) unit - digits of squares

[\begin{matrix} (i) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ (ii) & 0 & 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 \\ (iii) & 0 & 1 & 4 & 9 & 6 & 5 & 6 & 9 & 4 & 1 \end{matrix}]

i s s u m o f t w o s q u a r e n u m b e r s .

H e n c e i t s u n i t - d i g i t (8) i s m a d e u p

o f s u m o f u n i t - d i g i t s o f s q u a r e s

O n l y t w o p o s s i b i l i t i e s :

4 + 4 = 8 & 9 + 9 = 18

^{∙} 4 i s s q u a r e o f u n i t - d i g i t s : 2 & 8

^{∙} 9 i s s q u a r e o f u n i t - d i g i t s : 3 & 7

S e a r c h f o r t h e n u m b e r s i s

n a r r o w e n o u g h n o w : O n l y

n u m b e r s w i t h u n i t s 2, 8, 3 & 7

S o f i n a l l y 3 i s s u c c e s s f u l

c a n d i d a t e . I f n i s o n e p o s s i b l e

c a n d i d a t e f o r r e q u i r e d n u m b e r s

o t h e r n u m b e r N i s

N = \sqrt{2018 - n}

p r o v i d e d t h a t 2018 - n i s p r e f e c t

s q u r e .

F i n a l l y 13^{2} + 43^{2} = 2018 o n l y

s o l u t i o n .

Commented by floor(10²Eta[1]) last updated on 01/Sep/20

cool!! with that and knowing that

the numbers are both odd helps a lot

because we just have to look for

the numbers (3, 7, 13, 17, 23, 27, 33, 37, 43)

Commented by Rasheed.Sindhi last updated on 01/Sep/20

T han k s floor sir!