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carpetpool 2020-03-12 02:50

Orders of consecutive elements does not exceed floor(sqrt(p))
 
For some prime p, let m = ord[SUB]p[/SUB](2) be the multiplicative order of 2 mod p, and m[SUB]2[/SUB] = ord[SUB]p[/SUB](3) be the order of 3 mod p. Let L be the least common multiple of m and m[SUB]2[/SUB] (L = lcm(m,m[SUB]2[/SUB])).

Does a prime p exist such that L < sqrt(p) or simply floor(sqrt(p)) ?

(There is no such prime below 10^9)

The question in general is, for integers (a,b) (a ≠ b[SUP]i[/SUP] for some i > 2 or vice versa) are there finitely many primes p such that:

L > floor(sqrt(p))

where
L = lcm(m,m[SUB]2[/SUB])
m = ord[SUB]p[/SUB](a) and m[SUB]2[/SUB] = ord[SUB]p[/SUB](b) ?


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