Chapter 01 - Introduction.sagews

Chapter 1 - Introduction

The procedure below produces the table on page 2. See Chapter 5 for information about the mod operation.

P=primes_first_n(100001)
S=[]
T=[3]
tbl=[]
for m in [0..4]:
    for k in [10^m+1..10^(m+1)]:
        if mod(P[k],4)==1:
            S.append(P[k])
        else:
            T.append(P[k])
    tbl.append([len(S),len(T)])
table([(10^(m+1),tbl[m][0],tbl[m][1]) for m in [0..4]], header_row=["n", "primes of the form 4k+1","primes of the form 4k+3"], frame=True)
    

+--------+-------------------------+-------------------------+ | n | primes of the form 4k+1 | primes of the form 4k+3 | +========+=========================+=========================+ | 10 | 4 | 6 | +--------+-------------------------+-------------------------+ | 100 | 47 | 53 | +--------+-------------------------+-------------------------+ | 1000 | 495 | 505 | +--------+-------------------------+-------------------------+ | 10000 | 4984 | 5016 | +--------+-------------------------+-------------------------+ | 100000 | 49950 | 50050 | +--------+-------------------------+-------------------------+