Calculus How To

Sufficiently Large

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Loosely speaking, sufficiently large means “large enough” or “sufficiently large numbers”. In mathematics though, we want to define things a little more precisely. Exactly what makes a constant, term or other quantity “large enough” really depends where you’re using the expression. It could be very well defined (for example, a quantity greater than 10) or it could be an estimate. In some cases, it might be theoretically possible but not calculable.

Examples of Sufficiently Large

Weakly complete sequences: A weakly complete sequence is one where every “sufficiently large” natural number is a sum of a sequence’s terms [1]. In other words, it’s a sequence that doesn’t seem to be complete at first, but as you travel down the number line (i.e. as the numbers get “large enough”), the sequence meets the definition of completeness. There are an infinite number of possible sequences; What numbers are sufficiently large depends on the specific sequence.

Hardy-Littlewood conjecture: This famous theory states that every sufficiently large number (i.e. numbers beyond a certain point) can be expressed as a sum of a square and a prime and every large enough number is the sum of a cube and a prime. This theory was later dropped when Hooley [2] & Linnick [3] proved that a sufficiently large enough integer is the sum of two squares and a prime (assuming the extended Riemann hypothesis) [4]. The important thing here is that it happens at some point; the exact numerical value is largely irrelevant.

A Haken-manifold is manifold containing a properly embedded 2-sided incompressible surface; If a 3-manifold meets this property, it’s called sufficiently large [5].

sufficiently large

A 3-manifold.

In statistics, we’re often concerned with getting a sufficiently large sample: one that’s big enough to represent some aspect of the population (like the mean, for example). See: Large Enough Sample Condition (


[1] Fox, A. & Knapp, M. (2013). A Note on Weakly Complete Sequences. Journal of Integer Sequences.
[2] Hooley, C. (1957). On the representation of a number as the sum of two squares and a prime. Acta, Math. 97, 109-210.
[3] Linnick, J. (1960). An Asymptotic Formula in an Additive Problem of Hardy & Littlewood (Russian). Izv. Akad. Nauk SSR, Ser. Mat. 24. 629-706.
[4] Hardy, G. & Rao, M. Semi r-free and r-free integers; A Unified approach. Canadian Mathematical Bulletin (Sep, 1982).
[5] Waldhausen, F. (1968). On Irreducible 3-Manifolds Which are
Sufficiently Large. Annals of Mathematics, Second Series Volum 87 No. 1. Princeton University.

Stephanie Glen. "Sufficiently Large" From Calculus for the rest of us!

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