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NSW Number

An NSW number (named after Newman, Shanks, and Williams) is an integer that solves the Diophantine equation

(1)

In other words, the NSW numbers index the diagonals of squares of side length having the property that the squares of the diagonal equals one plus a square number . Such numbers were called "rational diagonals" by the Greeks (Wells 1986, p. 70). The name "NSW number" derives from the names of the authors of the paper on the subject written by Newman et al. (1980/81).

The first few NSW numbers are therefore , 7, 41, 239, 1393, ... (Sloane's A002315), which correspond to square side lengths , 5, 29, 169, 985, 5741, 33461, 195025, ... (Sloane's A001653). The values indexed by and therefore give 2, 50, 1682, 57122, ... (Sloane's A088920).

Taking twice the NSW numbers gives the sequence 2, 14, 82, 478, 2786, 16238, ... (Sloane's A077444), which is exactly every other Pell-Lucas number.

The first few prime NSW numbers are , 41, 239, 9369319, 63018038201, 489133282872437279, ... (Sloane's A088165), corresponding to indices , 2, 3, 9, 14, 23, 29, 81, 128, 210, 468, 473, 746, 950, 3344, 4043, 4839, 14376, 39521, 64563, 72984, 82899, 84338, 85206, 86121, ... (Sloane's A113501).

The following table summarizes the largest known NSW primes, where the indices correspond via to the indices of prime half-Pell-Lucas numbers that are odd.

decimal digits	discoverer	date
	E. W. Weisstein	May 19, 2006
	E. W. Weisstein	Aug. 29, 2006
	E. W. Weisstein	Nov. 16, 2006
	E. W. Weisstein	Nov. 26, 2006
	E. W. Weisstein	Dec. 10, 2006
	E. W. Weisstein	Jan. 25, 2007

Interestingly, the values give every other convergent to Pythagoras's constant .

Explicit formula for and are given by

			(2)
			(3)

for positive integers (Ribenboim 1996, p. 367). A recurrence relation for is given by

(4)

with and .

REFERENCES:

Newman, M.; Shanks, D.; and Williams, H. C. "Simple Groups of Square Order and an Interesting Sequence of Primes." Acta Arith. 38, 129-140, 1980/81.

Ribenboim, P. "The NSW Primes." §5.9 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 367-369, 1996.

Sloane, N. J. A. Sequences A001653/M3955, A002315/M4423, A077444, A088165, A088920, and A113501 in "The On-Line Encyclopedia of Integer Sequences."

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 70, 1986.

CITE THIS AS:

Weisstein, Eric W. "NSW Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NSWNumber.html