prime_gaps.sagews

def prime_g(B):
    """
    Compute the numbers g(p) for all primes p <= B, as a time series.

    These are the numbers g(p)=(next_prime(p)-p)/p, which are numbers between 0 and 1.
    """
    v = [float(p) for p in prime_range(next_prime(B)+1)]  # primes less than B and one more.
    return stats.TimeSeries([(v[i+1]-v[i])/v[i] for i in range(len(v)-1)])

%time t = prime_g(10^7); t

Error in lines 1-1 Traceback (most recent call last): File "/mnt/home/4by9x1bp/.sagemathcloud/sage_server.py", line 671, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "", line 8, in prime_g File "c_lib.pyx", line 70, in sage.ext.c_lib.sage_python_check_interrupt (sage/ext/c_lib.c:925) KeyboardInterrupt

CPU time: 7.02 s, Wall time: 6.91 s

%time show(t.plot_histogram(bins=5000), xmax=.01,ymax=150)

CPU time: 10.65 s, Wall time: 8.76 s

t2 = stats.TimeSeries([x for x in t if x<=.002])

len(t)
len(t2)

664579 663995

%time show(t2.plot_histogram(bins=1000), ymax=10000)

CPU time: 1.92 s, Wall time: 1.78 s

t3 = stats.TimeSeries([x for x in t2 if x<=.001])
len(t)
len(t2)
len(t3)
%time show(t3.plot_histogram(bins=1000), ymax=10000)

664579 663995 663432

CPU time: 1.94 s, Wall time: 1.81 s

%time t = prime_g(10^8); t

[0.5000, 0.6667, 0.4000, 0.5714, 0.1818 ... 0.0000, 0.0000, 0.0000, 0.0000, 0.0000] CPU time: 7.95 s, Wall time: 7.94 s

%time show(t.plot_histogram(bins=3000), xmax=.01,ymax=150)

CPU time: 5.07 s, Wall time: 4.98 s

t2 = stats.TimeSeries([x for x in t if x<=.001])
len(t)
len(t2)
%time show(t2.plot_histogram(bins=1000), ymax=10000)

5761455 5760308

CPU time: 2.24 s, Wall time: 2.12 s

%time show(t2.plot_histogram(bins=1000), ymax=20000, xmax=2e-4)

CPU time: 1.95 s, Wall time: 1.89 s

%time t = prime_g(10^9); t

[0.5000, 0.6667, 0.4000, 0.5714, 0.1818 ... 0.0000, 0.0000, 0.0000, 0.0000, 0.0000] CPU time: 75.75 s, Wall time: 75.64 s

%time show(t.plot_histogram(bins=3000), xmax=.01,ymax=150)

CPU time: 7.95 s, Wall time: 7.86 s

eps = float(0.001)
t2 = stats.TimeSeries([x for x in t if x<=eps])
len(t)
len(t2)
%time show(t2.plot_histogram(bins=1000), ymax=10000)

50847534 50846387

CPU time: 3.57 s, Wall time: 3.46 s

%time show(t2.plot_histogram(bins=1000), ymax=20000, xmax=5e-5)

CPU time: 2.80 s, Wall time: 2.63 s

%time show(t2.plot_histogram(bins=5000), ymax=20000, xmax=1e-5)

CPU time: 9.60 s, Wall time: 9.50 s