Find all natural numbers x and y satisfying

Rowan Callahan

Rowan Callahan

Answered question

2022-04-02

Find all natural numbers x and y satisfying the equality 19(x2+y2)=221+33xy

Answer & Explanation

paganizaxpo3

paganizaxpo3

Beginner2022-04-03Added 8 answers

Step 1
19(x33y38)2+(1919(3338)2)y2=221,
Step 2
which gives
(1919(3338)2)y2221
or 1y6.
The similar thing we can make for x. Now, check it.
disolutoxz61

disolutoxz61

Beginner2022-04-04Added 12 answers

A simpler approach:
19y2(33x)y+19x2221=0
we solve this for y:
Δ=(33x)24×19×(19x2221)
Or: Δ=167963550x2<47.31
that is numbers 36,25,16,9,4,1 can be checked. Only For x2=25 and x2=4 we have integers for Δ as 89 and 124 respectively which gives x=5,y=2. Since the equation is symmetric for x and y we also have x=2,y=5.

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