Shared2015-01-30-104847.sagewsOpen in CoCalc
Author: Paul Zeitz
Views : 4
3

3
p=11
a=8
sum =0
h=(p-1)/2
for k in [1..h]:
if k*a/p-floor(k*a/p)>1/2:
sum += 1
print legendre_symbol(a,p)==(-1)^sum

True
sum

3
int(mod(32,11))+3

13
mod(32,11)>5

True
legendre_symbol(5,29)

1
for k in [1..14]:
if mod(k^2,29)==5:
print k, k^2

11 121
173.is_prime()

True
p = 17
a = 2
val = 0
h = (p-1)/2
for k in [1..h]:
if mod(k^2,p) == a:
val = 1
print val

1
legendre_symbol(7,11)

-1
sigma(9)

13
sigma(100)

217
sigma(32)

63
def mif(f,n):
sum = 0
for d in n.divisors():
sum += moebius(d)*f(n/d)
return sum

mif(sigma,4)

4
def sq(n):
return n^2

mif(sq,7)

48
mif(sq,2)

3
mif(sq,75)

4800
p=107
a=6
sum=0
for k in [1..(p-1)]:
sum += legendre_symbol(k,p)
sum

0
133.is_prime()

False
d=210
l = 10
n=100
P=Primes()
for k in range(n):
val =P.unrank(k)
start=val
tVal = True
for m in [1..(l-1)]:
val += d
tVal *= val.is_prime()
if tVal:
print start

199
7
37
67
97
127
127.is_prime()

7 37 67 97 127 True
199+9*210

2089
2089.is_prime()

True
400-121

279
factor(279)

3^2 * 31
(109+1)/2

55
2*54*55

5940
5940^2+109^2

35295481
sqrt(35295481 )

5941
4892*11

53812
legendre_symbol(23,29)

1
n= 29*37

n

1073
x=100
y=mod(100^13,n)
y

618
mod(618^853,1073)

100