Identifying relation between numbers based on equation relating

Willow Cooper

Willow Cooper

Answered question

2022-04-08

Identifying relation between numbers based on equation relating them
If a2+b2+16c2=2(3ab+6bc+4ac), where a,b,c are non zero numbers. Then a,b,c are in ?

Answer & Explanation

Dubliddissefyf8

Dubliddissefyf8

Beginner2022-04-09Added 10 answers

There is a way to make this appear simpler. You are using letters a,b,c. Alright, take any x,y,z you like such that 2x216y2+81z2=0. This is just a point on a cone. Then still to avoid fractions, let a=4x+12y11z;,;;b=4y9z;,;;c=4z
For infinitely many examples, we can introduce variables u, v and take
x=576u2+72uv;,;;y=594u2+144uv+9v2;,;;z=248u2+64uv+4v2
For instance, let u=1,v=1 to get
x=648,y=747,z=316, then
a=8080,b=144,c=1264. These are all multiples of 16, we can divide out to get a=505,b=9,c=79.
Here is a good one, as a single step:
a=16u2+36uv+19v2;,;;b=9v2;,;;c=4u22v2
With u=1,v=1 we get a=19,b=9,c=2, or with u=1,v=1 we get a=71,b=9,c=2.
PTHP=D
(100310114941)(1343164616)(131140194001)=(10008000812)

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