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Monday, March 26, 2012

Andrica's Conjecture

Prime numbers are of course an unending source of fascinating mathematical work and conjecture. One prime number conundrum I learned of from Clifford Pickover's "The Math Book" is "Andrica's Conjecture" (after Dorin Andrica) which states that the square root of any nth prime number minus the square root of the (n-1)th prime will always result in a number less than 1; i.e. the difference of the square roots of any two consecutive primes will always be less than 1... \sqrt{p_{n+1}} - \sqrt{p_n} < 1

It has been successfully  tested out to beyond the 10^16th prime! In fact interestingly, the largest difference thus far comes with the 4th prime and equals only ≈ 0.670873. From there the values appear to trend asymptotically downward (thus, nowhere close to 1)… but still, the conjecture remains UNproven. More at Wikipedia here:
 

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