I just noticed that dividing 1÷998 gives me the apparently non-periodic
0.001002004008016032064…,
which is 10−3+2×10−6+4×10−9+8×10−12+⋯=∞∑i=02i10−3(i+1).
Since every rational number expands into a finite or periodic decimal expansion, does that become periodic somehow? If so, how?
Answer
It does repeat, but with a very long period: Factoring the denominator into primes gives
998=2⋅499;
since 2 is a factor of 10 and 10 is a primitive root modulo 499, the period of repetition is 499−1=498 digits long. Indeed, consulting WolframAlpha gives:
1998=0.¯00100200400801603206412825651302605210420841683366733466¯93386773547094188376753507014028056112224448897795591182¯36472945891783567134268537074148296593186372745490981963¯92785571142284569138276553106212424849699398797595190380¯76152304609218436873747494989979959919839679358717434869¯73947895791583166332665330661322645290581162324649298597¯19438877755511022044088176352705410821643286573146292585¯17034068136272545090180360721442885771543086172344689378¯7575150300601202404809619238476953907815631262525050.
(The is analogous to the repetition of the decimal digits of, e.g., 1/14: We have 14=2⋅7 and that 10 is a primitive root modulo 7, so the period of the decimal expansion of 1/14 is 7.)
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