a-2-b-2-c-2-2019-a-b-c-are-prime-numbers-how-many-possible-triples-of-a-b-c-which-that-suitable-for-equation-above-

Question Number 51950 by naka3546 last updated on 01/Jan/19

a^{2} + b^{2} + c^{2} = 2019

a, b, c a r e p r i m e n u m b e r s .

h o w m a n y p o s s i b l e t r i p l e s o f (a, b, c) w h i c h t h a t s u i t a b l e f o r e q u a t i o n a b o v e .

Answered by afachri last updated on 01/Jan/19

Commented by naka3546 last updated on 01/Jan/19

h o w t o g e t i t ?

Answered by mr W last updated on 02/Jan/19

s a y a ⩽ b ⩽ c

c ⩽ \sqrt{2019 - 2} \approx 45, i . e . t h e l a r g e s t p r i m e

c o u l d m a x . b e 43 .

\Rightarrow a, b, c \in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43)

t h e s q u a r e s o f t h e s e n u m b e r s a r e

n n^{2}

1 1

2 4

3 9

5 25

7 49

11 121

13 169

17 289

19 361

23 529

29 841

31 961

37 1369

41 1681

43 1849

w e s t a r t w i t h c

c = 43 :

a^{2} + b^{2} = 2019 - 1849 = 170

\Rightarrow b = 13, a = 1 ∙

\Rightarrow b = 11, a = 7 ∙

c = 41 :

a^{2} + b^{2} = 2019 - 1681 = 338

b = 17 \Rightarrow a^{2} = 338 - 289 = 49 \Rightarrow a = 7 ∙

b = 13 \Rightarrow a^{2} = 338 - 13^{2} = 169 \Rightarrow a = 13 ∙

b = 11 \Rightarrow a^{2} = 338 - 11^{2} = 217 \Rightarrow n o s o l . f o r a ⩽ 11

\Rightarrow n o f u r t h e r s o l u t i o n f o r b < 11

c = 37 :

a^{2} + b^{2} = 2019 - 1369 = 650

b = 23 \Rightarrow a^{2} = 650 - 529 = 121 \Rightarrow a = 11 ∙

b = 19 \Rightarrow a^{2} = 650 - 361 = 289 \Rightarrow a = 17 ∙

b = 17 \Rightarrow a^{2} = 650 - 289 = 361 \Rightarrow n o s o l . f o r a ⩽ 17

\Rightarrow n o f u r t h e r s o l u t i o n f o r b < 17

c = 31 :

a^{2} + b^{2} = 2019 - 961 = 1058

b = 31 \Rightarrow a^{2} = 1058 - 961 = 97 \Rightarrow n o s o l . f o r a ⩽ 31

b = 29 \Rightarrow a^{2} = 1058 - 841 = 217 \Rightarrow n o s o l . f o r a ⩽ 29

b = 23 \Rightarrow a^{2} = 1058 - 529 = 529 \Rightarrow a = 23 ∙

\Rightarrow n o f u r t h e r s o l u t i o n f o r b < 23

c = 29 :

a^{2} + b^{2} = 2019 - 841 = 1178

b = 29 \Rightarrow a^{2} = 1178 - 841 = 337 \Rightarrow n o s o l . f o r a ⩽ 29

b = 23 \Rightarrow a^{2} = 1178 - 529 = 649 \Rightarrow n o s o l . f o r a ⩽ 23

\Rightarrow n o f u r t h e r s o l u t i o n f o r b < 23

\Rightarrow n o f u r t h e r s o l u t i o n f o r c < 29

\Rightarrow a l l s o l u t i o n s a r e

(1, 13, 43)

(7, 11, 43)

(7, 17, 41)

(13, 13, 41)

(11, 23, 37)

(17, 19, 37)

(23, 23, 31)

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