Consecutive integers and relatively prime numbers

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SUMMARY

Consecutive integers are always relatively prime to each other, a fact that has been established since ancient times, likely known since the era of Euclid. The proof relies on the fundamental property of divisors, where if a divisor d divides both a and (a+1), it must also divide their difference, which is 1, leading to the conclusion that d can only be ±1. This elementary concept does not require extensive formal proof due to its straightforward nature.

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Summary:: Interested in the history of the proof.

Consecutive integer numbers are always relatively prime to each other. Does anyone know when this was proved? Was this known since Euclid's time or was this proved in modern times?
 
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I assume this has been known since someone observed divisors. It is so elementary that it doesn't even deserve the label 'proof': ##d|a \wedge d|(a+1) \Longrightarrow d|((a+1)-a)=1 \Longrightarrow d\in \{\pm 1\}.##

I don't think there is a book History Of Proofs. The earliest proofs are probably the geometry of the ancient Greeks.
 

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