Pronic number

A pronic number is a number that is the product of two consecutive integers, that is, a number of the form ${\displaystyle n(n+1)}$. ^[1] The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, ^[2] or rectangular numbers; ^[3] however, the term "rectangular number" has also been applied to the composite numbers. ^{[4] [5]}

The first few pronic numbers are:

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 … (sequence A002378 in the OEIS ).

Letting ${\displaystyle P_{n}}$ denote the pronic number ${\displaystyle n(n+1)}$, we have ${\displaystyle P_{{-}n}=P_{n{-}1}}$. Therefore, in discussing pronic numbers, we may assume that ${\displaystyle n\geq 0}$ without loss of generality, a convention that is adopted in the following sections.

As figurate numbers

Twice a triangular number is a pronic number

The nth pronic number is n more than the nth square number

The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's Metaphysics , ^[2] and their discovery has been attributed much earlier to the Pythagoreans. ^[3] As a kind of figurate number, the pronic numbers are sometimes called oblong ^[2] because they are analogous to polygonal numbers in this way: ^[1]

1 × 2	2 × 3	3 × 4	4 × 5

The nth pronic number is the sum of the first n even integers, and as such is twice the nth triangular number ^{[1] [2]} and n more than the nth square number, as given by the alternative formula n² + n for pronic numbers. Hence the nth pronic number and the nth square number (the sum of the first n odd integers) form a superparticular ratio:

${\displaystyle {\frac {n(n+1)}{n^{2}}}={\frac {n+1}{n}}}$

Due to this ratio, the nth pronic number is at a radius of n and n + 1 from a perfect square, and the nth perfect square is at a radius of n from a pronic number. The nth pronic number is also the difference between the odd square (2n + 1)² and the (n+1)st centered hexagonal number.

Since the number of off-diagonal entries in a square matrix is twice a triangular number, it is a pronic number. ^[6]

Sum of pronic numbers

The partial sum of the first n positive pronic numbers is twice the value of the nth tetrahedral number:

${\displaystyle \sum _{k=1}^{n}k(k+1)={\frac {n(n+1)(n+2)}{3}}=2T_{n}}$.

The sum of the reciprocals of the positive pronic numbers (excluding 0) is a telescoping series that sums to 1: ^[7]

${\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{20}}\cdots =1}$.

The partial sum of the first n terms in this series is ^[7]

${\displaystyle \sum _{i=1}^{n}{\frac {1}{i(i+1)}}={\frac {n}{n+1}}}$.

The alternating sum of the reciprocals of the positive pronic numbers (excluding 0) is a convergent series:

${\displaystyle \sum _{i=1}^{\infty }{\frac {(-1)^{i+1}}{i(i+1)}}={\frac {1}{2}}-{\frac {1}{6}}+{\frac {1}{12}}-{\frac {1}{20}}\cdots =\log(4)-1}$.

Additional properties

Pronic numbers are even, and 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number. ^{[8] [9]}

The arithmetic mean of two consecutive pronic numbers is a square number:

${\displaystyle {\frac {n(n+1)+(n+1)(n+2)}{2}}=(n+1)^{2}}$

So there is a square between any two consecutive pronic numbers. It is unique, since

${\displaystyle n^{2}\leq n(n+1)<(n+1)^{2}<(n+1)(n+2)<(n+2)^{2}.}$

Another consequence of this chain of inequalities is the following property. If m is a pronic number, then the following holds:

${\displaystyle \lfloor {\sqrt {m}}\rfloor \cdot \lceil {\sqrt {m}}\rceil =m.}$

The fact that consecutive integers are coprime and that a pronic number is the product of two consecutive integers leads to a number of properties. Each distinct prime factor of a pronic number is present in only one of the factors n or n + 1. Thus a pronic number is squarefree if and only if n and n + 1 are also squarefree. The number of distinct prime factors of a pronic number is the sum of the number of distinct prime factors of n and n + 1.

If 25 is appended to the decimal representation of any pronic number, the result is a square number, the square of a number ending on 5; for example, 625 = 25² and 1225 = 35². This is so because

${\displaystyle 100n(n+1)+25=100n^{2}+100n+25=(10n+5)^{2}}$.

The difference between two consecutive unit fractions is the reciprocal of a pronic number: ^[10]

${\displaystyle {\frac {1}{n}}-{\frac {1}{n+1}}={\frac {(n+1)-n}{n(n+1)}}={\frac {1}{n(n+1)}}}$

References

1. Conway, J. H.; Guy, R. K. (1996), The Book of Numbers, New York: Copernicus, Figure 2.15, p. 34.
2. Knorr, Wilbur Richard (1975), The evolution of the Euclidean elements, Dordrecht-Boston, Mass.: D. Reidel Publishing Co., pp. 144–150, ISBN   90-277-0509-7, MR   0472300 .
3. Ben-Menahem, Ari (2009), Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1, Springer reference, Springer-Verlag, p. 161, ISBN   9783540688310 .
4. "Plutarch, De Iside et Osiride, section 42", www.perseus.tufts.edu, retrieved 16 April 2018
5. Higgins, Peter Michael (2008), Number Story: From Counting to Cryptography, Copernicus Books, p. 9, ISBN   9781848000018 .
6. Rummel, Rudolf J. (1988), Applied Factor Analysis, Northwestern University Press, p. 319, ISBN   9780810108240 .
7. Frantz, Marc (2010), "The telescoping series in perspective", in Diefenderfer, Caren L.; Nelsen, Roger B. (eds.), The Calculus Collection: A Resource for AP and Beyond, Classroom Resource Materials, Mathematical Association of America, pp. 467–468, ISBN   9780883857618 .
8. McDaniel, Wayne L. (1998), "Pronic Lucas numbers" (PDF), Fibonacci Quarterly , 36 (1): 60–62, doi:10.1080/00150517.1998.12428962, MR   1605345, archived from the original (PDF) on 2017-07-05, retrieved 2011-05-21.
9. McDaniel, Wayne L. (1998), "Pronic Fibonacci numbers" (PDF), Fibonacci Quarterly , 36 (1): 56–59, doi:10.1080/00150517.1998.12428961, MR   1605341 .
10. Meyer, David. "A Useful Mathematical Trick, Telescoping Series, and the Infinite Sum of the Reciprocals of the Triangular Numbers" (PDF). David Meyer's GitHub. p. 1. Retrieved 2024-11-26.