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NSW Number

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An NSW number is a positive integer m for which the Diophantine equation

2n^2=m^2+1
(1)

has a positive integer solution n. Geometrically, a square of side length n has diagonal d=sqrt(2)n, whose square d^2 is one more than the square number m^2. Such diagonals 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 m=1, 7, 41, 239, 1393, ... (OEIS A002315), which correspond to square side lengths n=1, 5, 29, 169, 985, 5741, 33461, 195025, ... (OEIS A001653). The squared diagonal lengths d^2 are 2, 50, 1682, 57122, ... (OEIS A088920).

Taking twice the NSW numbers gives the sequence 2, 14, 82, 478, 2786, 16238, ... (OEIS A077444), which is exactly the odd-indexed Pell-Lucas numbers.

The first few prime NSW numbers are m=7, 41, 239, 9369319, 63018038201, 489133282872437279, ... (OEIS A088165). The reported indices of prime or probable prime NSW numbers are k=1, 2, 3, 9, 14, 23, 29, 81, 128, 210, 468, 473, 746, 950, 3344, 4043, 4839, 14376, 39521, 64563, 72984, 82899, 84338, 85206, 86121, 139160, 276141, ... (OEIS A113501; OEIS A099088, using k^'=2k+1).

The following table summarizes NSW prime number and probable prime records reported through 2023, where the indices k correspond via k=(k^'-1)/2 to the odd indices k^' of prime or probable prime half-Pell-Lucas numbers.

kdecimal digitsdiscovererdate
6456349427E. W. WeissteinMay 19, 2006
7298455874E. W. WeissteinAug. 29, 2006
8289963464E. W. WeissteinNov. 16, 2006
8433864566E. W. WeissteinNov. 26, 2006
8520665230E. W. WeissteinDec. 10, 2006
8612165931E. W. WeissteinJan. 25, 2007
139160106535R. PriceDec. 7, 2018
276141211401R. Pricereported Dec. 2023

R. Price reported the probable prime NSW number with index 276141 in Dec. 2023 (OEIS A099088). It equals Q_(552283)/2, where Q_n is the nth Pell-Lucas number, and has 211,401 decimal digits.

Interestingly, the values m/n give every other convergent to Pythagoras's constant sqrt(2).

Explicit formulas for m and n are

m=((1+sqrt(2))^(2k+1)+(1-sqrt(2))^(2k+1))/2
(2)
n=((2+sqrt(2))^(2k+1)+(2-sqrt(2))^(2k+1))/(2^(k+2))
(3)

for nonnegative integers k (Ribenboim 1996, p. 367, with the index shifted by one). A recurrence relation for m=S(k) is given by

S(k)=6S(k-1)-S(k-2)
(4)

for k>=2, with S(0)=1 and S(1)=7.

See also

Integer Sequence Primes, Pell-Lucas Number, Pell Number, Probable Prime, Pythagoras's Constant

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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, A099088, 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.

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NSW Number

Cite this as:

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

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