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Why do these integers give rise to constants when repeatedly subtracted from one another?

\begin{array}{r|cccccccc} & 1 && 2 && 3 && 4 && 5 && 6 && 7 \\ \hline \text{squares} & 1 && 4 && 9 && 16 && 25 && 36 && 49 \\ \text{differences} & & 3 && 5 && 7 && 9 && 11 && 13 \\ \text{constant} &&& 2 && 2 && 2 && 2 && 2 \\ \hline \text{cubes} & 1 && 8 && 27 && 64 && 125 && 216 && 343 \\ \text{differences} && 7 && 19 && 37 && 61 && 91 && 127 \\ \text{differences} &&& 12 && 18 && 24 && 30 && 36 \\ \text{constant} &&&& 6 && 6 && 6 && 6 \\ \hline \text{powers of 4} & 1 && 16 && 81 && 256 && 625 && 1296 && 2401 \\ \text{differences} && 15 && 65 && 175 && 369 && 671 && 1105 \\ \text{differences} &&& 50 && 110 && 194 && 302 && 434 \\ \text{differences} &&&& 60 && 84 && 108 && 132 && 156 && 180 \\ \text{constant} &&&&& 24 && 24 && 24 && 24 \\ \hline \end{array}

Blue
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  • with squares, difference is 2, with cubes , difference is 3x2=6, with 4th-powers, difference is 4x3x2=24, with 5th-powers, difference will be 5x4x3x2=120, etc. – Lourrran Jun 25 '25 at 13:53
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    Welcome to Math.SE! ... Congratulations on your observation! ... The term-of-art here is "Finite Differences", sometimes called the "finite difference calculus". The Wikipedia entry is a bit advanced, but now you have a target for searching the web for a beginner-friendly discussion. One upshot is that when a sequence is governed by any polynomial (not just simple powers), the iterated differences always eventually become constant; moreover, one can recover the coefficients of the poly from the calculated diffs. It's neat stuff. ... Cheers! – Blue Jun 25 '25 at 13:59
  • Conway and Guy's Book of Numbers has some interesting material on nyumber patterns of this and other kinds – Mark Bennet Jun 25 '25 at 14:24

2 Answers2

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Differencing is a discrete analgoue of differentiation. When we take derivatives of polynomials, the power goes down by $1$ each time, and hence, eventually you with a constant. In general for any series of power $n$, after $n-1$ differencing steps, you will end up with the constant $n!$ (just like $n-1$ derivatives).

Note if you did the same thing with an infinite polynomial (i.e., Taylor series) like $\sin(x)$ or $e^x$, the differences you would never reach a constant.

Pavan C.
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Here's part of what you're looking for - $(x+1)^2 - x^2 = 2x+1$. Can you extend that pattern?

Bo Rel
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