Algorithms

NeverHeardOfMe wrote:

If p is prime, then p ^ 2 = 1 + 24 * n, for some n

Don't know about you, but I think that's quite amazing. Why is it true at all, and given that it is, where does 24 come from?! (Pity it's not 42 really!)

p^6 = 1 + 42n for some n

p^6 = 1 + 84n for some n, and p being a prime >7

p^3

(p^6 - 1) = (p^3 - 1) * (p^3 + 1) = 2*a * 2*b = 4*a*b

p^3

(p^6 - 1) = (p^3 - 1) * (p^3 + 1) = 3*c * d

p = (7*o + e) with o being an odd integer and e an even integer in {2; 4; 6; 8; 10; 12}

p^3 = (7*o + e)^3 = (7*o)^3 + 3*e*(7*o)^2 + 3*(7*o)*e^2 + e^3

- e=2:  e^3=   8 => -1
- e=4:  e^3=  64 => -1
- e=6:  e^3= 216 => +1
- e=8:  e^3= 512 => +1
- e=10: e^3=1000 => +1
- e=12: e^3=1728 => +1

(p^6 - 1) = (p^3 - 1) * (p^3 + 1) = 7*e * f

(p^6 - 1) = (p^3 - 1) * (p^3 + 1) = 4*3*7*n

p^6 = 1 + 84n

(p^6 - 1) = (p^3 - 1) * (p^3 + 1) = 2*a * 4*b = 8*a*b

p^6 = 1 + 168n

(p^6 - 1) = 9*c

p^6 = 1 + 504n

xNew=x*cos(alpha)+y*sin(alpha);
yNew=-x*sin)alpha)+y*cos(alpha);

widNew=wid*abs(cos(alpha))+hei*abs(sin(alpha));
heiNew=wid*abs(sin(alpha))+hei*abs(cos(alpha));

new_width  = abs( width * cos(alpha) + height * sin(alpha))
new_height = abs(-width * sin(alpha) + height * cos(alpha))

rotated_x =  x * cos(alpha) + y * sin(alpha)
rotated_y = -x * sin(alpha) + y * cos(alpha)

new_height = abs(-width * sin(alpha) + height * cos(alpha))

abs(-width * sin(alpha)) + abs(height * cos(alpha))
    ^
    |
    +- This is the minus that you are talking about, I suppose

abs(-x * sqrt(2) / 2 + x * sqrt(2) / 2)

abs(-x * a + x * a)

abs(0)

for x = 0 to xmax
  for y = 0 to ymax
    if (CI(x,y) > CI(x-1,y) || CI(x,y) > CI(x+1,y) || CI(x,y) > CI(x,y-1) || CI(x,y) > CI(x,y+1))
      set_line_colour(x,y);

int CI(int x, int y) 
{
  return (int)(altitude[x,y] / contour_interval);
}