- #1

Gear300

- 1,193

- 5

Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form

## P(x,y) = a x + b y + d xy ##

can map the integers onto the integers. So through a change of basis, I re-express this as

## P'(u,v) = Au^2 + Bv^2 ##

for rational A and B. ##u,v## can be made to cover integer x,y for rational values, so the problem reduces to whether or not P'(u,v) can map the rationals onto the integers. I say no.

## P(x,y) = a x + b y + d xy ##

can map the integers onto the integers. So through a change of basis, I re-express this as

## P'(u,v) = Au^2 + Bv^2 ##

for rational A and B. ##u,v## can be made to cover integer x,y for rational values, so the problem reduces to whether or not P'(u,v) can map the rationals onto the integers. I say no.

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