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First Published January 2008

For some related ideas about quadratic intervals, perfect squares, and prime numbers, see: Biquadratic Paired Primes and Q-Paired Primes.

For this discussion, it is simply the interval between one perfect square on the number line and the next perfect square:

n*n,
(n+1)*(n+1),
(n+2)*(n+2),
(n+3)*(n+3),
. . . .

(For lists of perfect squares up to 1 trillion, see here.)

Of course, perfect squares are perfectly consistent. Their distribution is completely predictable because the interval between one perfect square and the next perfect square conforms to quadratic growth, meaning there exists a finite difference of the 2nd degree in the interval between each successive perfect square. Expressed as a polynomial, this is:

n^2 + 0
(or, simply, n^2)

The finite difference is 2nd degree because it appears in the second row of differences (not because the difference happens to be 2), as illustrated:

9    16    25    36    49
7   9   11   13

2    2    2

In terms of relating one perfect square to the next, the following formula could be given:

n = c+(c-l)+2

where n=next perfect square, c=current perfect square, l=last perfect square.

The concept of graphing numbers according to perfect square intervals originated in the Sacks Number Spiral. The term rotation is synonymous with quadratic interval.

Graphing numbers according to higher-degree polynomials, such as cubic (3rd degree), quartic (4th), and quintic (5th) should also yield interesting results.