# pronic

## English

### Etymology

Apparently from New Latin pronicus, a misspelling of Latin promicus, from Ancient Greek προμήκης (promḗkēs, elongated), but the spelling has been pronic from its earliest known occurrence in English ((Can we date this quote?) Leonhard Euler, Opera Omnia, series 1, volume 15).[1][2]

pronic (not comparable)

1. (mathematics) Of a number which is the product of two consecutive integers
• 1478 - Pierpaolo Muscharello, Algorismus p.163.[3]
Pronic root is as if you say, 9 times 9 makes 81. And now take the root of 9, which is 3, and this 3 is added above 81: it makes 84, so that the pronic root of 84 is said to be 3.
• 1794 - David Wilkie, Theory of interest, p.6, Edinburgh: Peter Hill, 1794.
When a = 2, and d = 2 also, in this case, in equation 1st, s=n2 + n = a pronic number, which is produced by the addition of even numbers in an arithmetic progression beginning at 2; and the pronic root ${\displaystyle \scriptstyle n={\frac {{\sqrt {4s+1}}-1}{2}}}$.
• 1804 - Paul Deighan, "Recommendatory letters", A complete treatise on arithmetic, rational and practical, vol.1, p.viii, Dublin: J. Jones, 1804.
As I admire each proposition fair,
the pronic number and the perfect square,
the puzzling intricate equation solv'd,
as Grecia's chief the Gordian knot dissolv'd;
- John Bartley
• 1814 - Charles Butler, Easy Introduction to Mathematics, p.96, Barlett & Newman, 1814
A pronic number is that which is equal to the sum of a square number and its root. Thus, 6, 12, 20, 30, &c. are pronic numbers.
• 1998 - Wayne L. McDaniel, "Pronic Lucas Numbers", The Fibonacci Quarterly, pp.60-62, 1998.
It may be noted that if Ln is a pronic number, then Ln is two times a triangular number.
• 2005 - G. K. Panda1 and P. K. Ray, "Cobalancing numbers and cobalancers", International Journal of Mathematics and Mathematical Sciences, vol.2005, iss.8, pp.1189-1200.
Thus, our search for cobalancing number is confined to the pronic triangular numbers, that is, triangular numbers that are also pronic numbers.

### References

1. ^ David J. Darling, The universal book of mathematics: from Abracadabra to Zeno's paradoxes, pp.257-258, John Wiley and Sons, 2004 →ISBN.
2. ^ "A002378: Oblong (or promic, pronic, or heteromecic) numbers: n(n+1)", Online Encyclopedia of Integer Sequences, accessed and archived 21 May 2011.
3. ^ Jens Høyrup, "What did the abbacus teachers aim at when they (sometimes) ended up doing mathematics?", New perspectives on mathematical practices, pp.47-75, World Scientific, 2009 →ISBN.