The conversation between human and AI assistant.
Is there any idea, how to prove (no computer) that 253*Sqrt(2) +874*Sqrt(3) +582*Sqrt(5) >3173?
Then?
Since Sqrt(2), Sqrt(3), Sqrt(5), and Sqrt(10) are irrational, they must be linearly independent over the rational numbers. Therefore, the coefficients of these irrational numbers in the left-hand side of the inequality must be rational multiples of one another. But a quick check shows that they are not. Hence, we have a contradiction, and the assumption is false. Therefore, 253*Sqrt(2) +874*Sqrt(3) +582*Sqrt(5) > 3173.