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Let us put reasonable bounds on the value of the world's population in AD 1. For the sake of example, I'll choose a lower bound of 50,000,000 (8 significant figures specified), and an upper bound of 200,000,000 (9 sig. figs. specified). Similarly, I'll put bounds on the population in AD 1000 at 250,000,000 and 350,000,000 (both exact).
Note that the world's population must be an integer greater than or equal to 0; therefore, these values are exact -- there can be no loss of precision in the calculation due to these numbers. Using x = 2 (exact) and a generation length of (exactly) 40 years, we generate the following table :
Population
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AD 1 | AD 1000 | C (rounded for convenience)
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50 mil | 250 mil | 1.0665
200 mil | 250 mil | 1.00897 (min)
50 mil | 350 mil | 1.0809 (max)
200 mil | 350 mil | 1.02264
P(n) is a monotonically increasing function of c for the values of x and n chosen. Therefore, if the actual populations in AD 1 and AD 1000 lie within the given bounds, then c must lie within the interval [1.00897, 1.0809].
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