When will the decimals in a/b repeat?

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The previous post looked at how many digits are in the reduced fraction for the nth harmonic number. I was curious about how long the cycle of digits in a harmonic number might be.

I wrote about the period length for the digits of fractions almost a decade ago. This post includes code so I can apply it to harmonic demoninators.

from sympy import lcm, factorint, n_order

def period(n):<br>factors = factorint(n)<br>exp2 = factors.get(2, 0)<br>exp5 = factors.get(5, 0)<br>r = max(exp2, exp5)

d = n // (2**exp2 * 5**exp5)<br>s = 1 if d == 1 else n_order(10, d)<br>return (r, s)

This function returns two numbers: r is the number of non-repeating digits at the beginning and s is the length of the repeating part.

The following code

from functools import reduce

def lcm_range(n):<br>return reduce(lcm, range(1, n + 1))

print( period( lcm_range(50) ) )

prints (5, 1275120) meaning that 1/lcm(1, 2, 3, …, 49, 50) has five non-repeating digits following by 1,275,120 digits that repeat ad infinitum. And so the decimals in the expansion of H50 go have a cycle length of 1,275,120.

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John D. Cook, PhD

My colleagues and I have decades of consulting experience helping companies solve complex problems involving data privacy, applied math, and statistics.

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