0

What are the integers $a$ and $b$ if $a\sqrt2 + b\sqrt3 = 0$?

I suppose answer should be $(a, b) = (0, 0)$. But I am unable to justify it.

Generalisation: What are integers $a_i$'s if $$ a_1\sqrt{p_1} + a_2\sqrt{p_2} + \cdots + a_n\sqrt{p_n} = 0, $$ where $p_i$'s are distinct primes?

In fact, if we replace prime $p_i$'s with any other distinct positive integers that are not perfect squares, would the values of $a_i$'s be same? And why?

Please help. Thanks in advance!

prashant sharma
  • 556
  • Can you show that $\sqrt{2/3}$ is irrational? That is the same question. Use "unique prime factorization" in your proof. Do the case $a\sqrt2 + b\sqrt3$ before attempting the generalization. – GEdgar May 13 '23 at 23:53
  • For the generalization to those that are not perfect squares, it is not true. Consider 2 and 8. – Ja_1941 May 14 '23 at 00:15

0 Answers0