n=5
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$\frac{4}{n}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ where $a,b,c\in\mathbb{N}$ such that $1\leq a\leq b\leq c$.
$\frac{4}{3n}\leq\frac{1}{a}<\frac{4}{n}$, so $\frac{n}{4}<a\leq\frac{3n}{4}$
$\frac{\frac{4}{n}-\frac{1}{a}}{2}\leq\frac{1}{b}<\frac{4}{n}-\frac{1}{a}$, so $\frac{1}{\frac{4}{n}-\frac{1}{a}}<b\leq\frac{2}{\frac{4}{n}-\frac{1}{a}}$
$\Rightarrow na<(4a-n)b\leq 2na$
$\frac{1}{c}=\frac{4}{n}-\frac{1}{a}-\frac{1}{b}$, so $c=\frac{1}{\frac{4}{n}-\frac{1}{a}-\frac{1}{b}} \Rightarrow c=\frac{nab}{4ab-nb-na}$
def count_erdosstraus(n):
count = 0
for a in range(floor(n/4)+1,floor(3*n/4)+1):
if (4/n-1/a)!=0:
for b in range(max(a,floor(1/(4/n-1/a))+1),floor(2/(4/n-1/a))+1):
if (4/n-1/a-1/b)!=0:
if 1/(4/n-1/a-1/b)>=b and 1/(4/n-1/a-1/b)==int(1/(4/n-1/a-1/b)):
count += 1
#print('$\\frac{4}{'+str(n)+'}=\\frac{1}{'+str(a)+'}+\\frac{1}{'+str(b)+'}+\\frac{1}{'+str(1/(4/n-1/a-1/b))+'}$')
return count
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import numpy as np
def count_erdosstraus_fast(n):
grid_a,grid_b = np.mgrid[floor(n/4)+1:floor(3*n/4)+1,floor(1/(4/n-1/(floor(3*n/4)+1)))+1:floor(2/(4/n-1/(floor(n/4)+1)))+1]
print(grid_a.shape)
invalid = ( (grid_a<=grid_b)*(n*grid_a < (4*grid_a-n)*grid_b)*((4*grid_a-n)*grid_b <= 2*n*grid_a) ).astype('int')
#grid_c = invalid*np.nan_to_num(n*grid_a*grid_b/(4*grid_a*grid_b-n*(grid_a+grid_b)))
grid_remainders = np.mod(n*grid_a*grid_b,4*grid_a*grid_b-n*(grid_a+grid_b))
invalid = invalid*((4*grid_a*grid_b-n*(grid_a+grid_b))>0)
count = np.sum((invalid*(grid_remainders==0)).astype('int'))
return count
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import numpy as np
denom = 5
def count_erdosstraus_fast_by_row(n):
a_bounds = [floor(n/denom)+1,floor(3*n/denom)+1]
count = 0
for a in range(a_bounds[0],a_bounds[1]):
b_bounds = [max(a,floor(1/(denom/n-1/a))+1),floor(2/(denom/n-1/a))+1]
line_b = np.arange(b_bounds[0],b_bounds[1])
line_remainders = np.mod(n*a*line_b,denom*a*line_b-n*(a+line_b))
count += np.sum(line_remainders==0)
return count
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points = []
oeis = []
for i in range(2,1500):
if i%100==0:
print(i)
points.append([i,count_erdosstraus_fast_by_row(i)])
if points[len(points)-1][1] == 0:
print(str(i)+': ' +str(points[len(points)-1][1]))
list_plot(points)
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 
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import colorsys
lists_mod_k = []
k = 30
for i in range(0,k):
lists_mod_k.append([])
for i in range(len(points)):
if is_prime(points[i][0]):
for j in range(1,k):
if points[i][0]%k == j:
lists_mod_k[j].append([points[i][0],points[i][1]])
#print(mod_four_is_one)
print(lists_mod_k[2])
A = scatter_plot([], facecolor='black')
B = scatter_plot([], facecolor='black')
for i in range(1,k):
rgb = colorsys.hsv_to_rgb(1/k*i, 1, 0.8)
A = A+scatter_plot(lists_mod_k[i], facecolor=rgb, edgecolor=rgb)
'''
for i in range(2,k,3):
rgb = colorsys.hsv_to_rgb(1/k*i, 1, 0.8)
B = B+scatter_plot(lists_mod_k[i], facecolor=rgb, edgecolor=rgb)
'''
show(A)
#show(B)
[[2, 1]] [[2, 1]] 
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#list_plot(mod_four_is_three, color='red')
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