 # nt.quantity concept – Does each geometric sequence acquire diminutive remainders modulo sizable prime? Answer

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nt.quantity concept – Does each geometric sequence acquire diminutive remainders modulo sizable prime?

The require query I’m all in favour of is the next.

Fix a diminutive $$varepsilonin(0,1)$$ and an integer $$qge 2$$ (chances are you’ll occupy that $$q$$ is prime if it helps although I consider it should not signify too mighty). For a sizable prime $$P$$ and an integer $$ainmathbb Z$$, outline $$G(a,P)={aq^mmod P: m=0,1,2,dots}$$ the place the remainders are taken within the meander $$(-P/2, P/2)$$ (i.e., with the minimal workable absolute worth).

Is it undoubted that for all primes $$P$$ exterior of a clique of density at most $$varepsilon$$ (in any sense of the phrase “density” that’s subadditive), $$G(a,P)$$ accommodates a rest within the meander $$(-varepsilon P,varepsilon P)$$ for each altenative of $$ain mathbb Z$$?

However I’ll breathe too all in favour of any nontrivial leads to the identical route plane in the event that they fall considerably in need of a whole respond (breathe it affirmative or traverse).

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