000RM.dvi

616 Strings of prime numbers

Appendix: The number spiral

Beginning with the origin, we trace out a spiral clockwise through the
lattice points. Along with this, we label the lattice points 0, 1, 2,... consecutively.

0 1

2

4

6

9

12

16

(^2025)
30
Given a positive integerN, let(2m−1)^2 be the largestoddsquare
≤N, and write
N=(2m−1)^2 +q, 0 ≤q< 8 m.
Then the numberNappears at the lattice point
⎧
⎪⎪
⎪⎨
⎪⎪
⎪⎩
(m,−m+q+1) ifq≤ 2 m− 1 ,
(3m−q− 1 ,m)if2m− 1 ≤q≤ 4 m− 1 ,
(−m, 5 m−q−1) if 4m− 1 ≤q≤ 6 m− 1 ,
(− 7 m+q+1,−m)if6m− 1 ≤q≤ 8 m− 1.
Denote byf(m, n)the number at the lattice point(m, n).
It is clear that along the 45-degree line,f(n, n)=2n(2n−1). Also,
f(−n, n)=4n^2 ifn≥ 0 ,
and
f(n,−(n−1)) = (2n−1)^2 ifn> 0.
More generally,
f(m, n)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

4 m^2 − 3 m+n ifm>|n|,

4 m^2 −m−n if −m=|m|>|n|,

4 n^2 −n−m ifn>|m|,

4 n^2 − 3 n+m if −n=|n|>|m|.