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The problem: https://projecteuler.net/problem=44

If we put $f(x) = \frac 1 2 x(3x-1)$, then $f'(x) = \frac 1 2 (6x - 1)$, which is positive for all $x \geq 1$. Thus, $P_n$ is an increasing sequence.

Another thing is that $P_n$ is an integer for each positive integer $n$. If $n$ is even, then $n(3n-1)$ is also even because it's $n$ times something. If $n$ is odd, then so is $3n$, which gives $3n-1$ even. That gives $n(3n-1)$ even for all positive integers $n$.

Using the quadratic formula, we find that if a positive number $A$ is a pentagonal number $P_n$, then it must be true that $6n - 1 = \sqrt{1 + 24A}$. That is $\sqrt{1 + 24A}$ is an integer and it is congruent to 5 modulo 6.

We now try to find a solution $i, j, m, n$ such that $P_i = P_n + P_m$ and $P_j = P_n - P_m$.

In [47]:
def P(n):
    return n*(3*n - 1)//2

def integer_sqrt(n):
    if n < 0:
        return -1
    if n == 1 or n == 0:
        return n
    lo = 1
    hi = n
    while lo <= hi:
        mid = (hi+lo)//2
        square = mid*mid
        if square == n:
            return mid
        elif square > n:
            hi = mid-1
        else:
            lo = mid+1
    return -1

def find_pentagonal_index(A):
    d = 1 + 24*A
    six_n_minus_1 = integer_sqrt(d)
    if six_n_minus_1 < 0:
        return 0
    if six_n_minus_1 % 6 != 5:
        return 0
    return (six_n_minus_1 + 1)//6

def search_for_first_solution():
    m = 1
    while True:
        P_m = P(m)
        for n in range(1, m):
            P_n = P(n)
            s = P_n + P_m
            d = P_m - P_n
            i = find_pentagonal_index(s)
            j = find_pentagonal_index(d)
            if i > 0 and j > 0:
                return (m, n, i, j)
        m += 1

(m0, n0, i0, j0) = search_for_first_solution()
print(m0, n0)
print(P(m0), P(n0))
print(i0, j0)
print(P(i0), P(j0))

2167 1020
7042750 1560090
2395 1912
8602840 5482660

Ok. So that took a while. The idea now is that we know that our minimized difference must be at most $M = 5482660$.

We have $P_{n+1} - P_n = 3n + 1$. Not just that, for two positive integers $n, k$, we have $P_{n+k} - P_n = \frac 1 2 \left (6k n + 3k^2 - k \right )$. That is, if we fix $k$, $P_{n+k} - P_n$ is an increasing afine function of $n$. This means we can easily bound our loop variables since we know $P_{n+k} - P_n \leq M.$ In summary, we need $n \leq \frac {2M + k - 3k^2} {6k}$ and $2M + k - 3k^2 > 0$.

The inequality $2M + k - 3k^2 > 0$, by using the quadratic formula, implies $k < \frac {1 + \sqrt{1 + 24M}} {6}$ and we know, since $M = P(1912)$, by the considerations above, that $\frac {1 + \sqrt{1 + 24M}} {6}$ is an integer.

In our search for better and better differences, once we find a better one, we can update how we calculate the upper bounds for k and n.

In [63]:
k = 1
M = P(j0)
k_upper = (1 + integer_sqrt(1 + 24*M))//6

least_m = m0
least_n = n0
least_i = i0
least_j = i0
least_D = M

import timeit

start_time = timeit.default_timer()

while k < k_upper:
    n_upper = (2*least_D + k - 3*k*k) // (6*k) + 1
    print(k, k_upper, n_upper, end = ';\t')
    for n in range(1, n_upper):
        P_n = P(n)
        m = n + k
        P_m = P(m)
        s = P_n + P_m
        d = P_m - P_n
        if d < least_D:
            i = find_pentagonal_index(s)
            j = find_pentagonal_index(d)
            if i > 0 and j > 0:
                least_m = m
                least_n = n
                least_i = i
                least_j = j
                least_D = d
                k_upper = (1 + integer_sqrt(1 + 24*least_D))//6
    k += 1

elapsed = timeit.default_timer() - start_time

print("Answer (m, n, i, j, D)", least_m, least_n, least_i, least_j, least_D)
print("It took", elapsed, "seconds.")

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636;	1380 1912 635;	1381 1912 634;	1382 1912 632;	1383 1912 631;	1384 1912 629;	1385 1912 628;	1386 1912 626;	1387 1912 625;	1388 1912 623;	1389 1912 622;	1390 1912 620;	1391 1912 619;	1392 1912 618;	1393 1912 616;	1394 1912 615;	1395 1912 613;	1396 1912 612;	1397 1912 610;	1398 1912 609;	1399 1912 607;	1400 1912 606;	1401 1912 605;	1402 1912 603;	1403 1912 602;	1404 1912 600;	1405 1912 599;	1406 1912 597;	1407 1912 596;	1408 1912 595;	1409 1912 593;	1410 1912 592;	1411 1912 590;	1412 1912 589;	1413 1912 588;	1414 1912 586;	1415 1912 585;	1416 1912 583;	1417 1912 582;	1418 1912 580;	1419 1912 579;	1420 1912 578;	1421 1912 576;	1422 1912 575;	1423 1912 573;	1424 1912 572;	1425 1912 571;	1426 1912 569;	1427 1912 568;	1428 1912 566;	1429 1912 565;	1430 1912 564;	1431 1912 562;	1432 1912 561;	1433 1912 560;	1434 1912 558;	1435 1912 557;	1436 1912 555;	1437 1912 554;	1438 1912 553;	1439 1912 551;	1440 1912 550;	1441 1912 548;	1442 1912 547;	1443 1912 546;	1444 1912 544;	1445 1912 543;	1446 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1912 67;	1848 1912 66;	1849 1912 65;	1850 1912 64;	1851 1912 62;	1852 1912 61;	1853 1912 60;	1854 1912 59;	1855 1912 58;	1856 1912 57;	1857 1912 56;	1858 1912 55;	1859 1912 54;	1860 1912 53;	1861 1912 52;	1862 1912 51;	1863 1912 50;	1864 1912 49;	1865 1912 48;	1866 1912 47;	1867 1912 46;	1868 1912 45;	1869 1912 44;	1870 1912 43;	1871 1912 42;	1872 1912 41;	1873 1912 40;	1874 1912 39;	1875 1912 38;	1876 1912 37;	1877 1912 36;	1878 1912 35;	1879 1912 34;	1880 1912 33;	1881 1912 32;	1882 1912 31;	1883 1912 30;	1884 1912 29;	1885 1912 28;	1886 1912 27;	1887 1912 26;	1888 1912 25;	1889 1912 24;	1890 1912 23;	1891 1912 22;	1892 1912 21;	1893 1912 20;	1894 1912 19;	1895 1912 18;	1896 1912 17;	1897 1912 16;	1898 1912 15;	1899 1912 14;	1900 1912 13;	1901 1912 12;	1902 1912 11;	1903 1912 10;	1904 1912 9;	1905 1912 8;	1906 1912 7;	1907 1912 6;	1908 1912 5;	1909 1912 4;	1910 1912 3;	1911 1912 2;	Answer (m, n, i, j, D) 2167 1020 2395 2395 5482660
It took 222.53819137000028 seconds.

So, we found our answer: $5482660$.

The computation wasn't fast. I liked how we could improve the upper limit on looping for k as we found better and better values for least_D. However, k_upper was never udpated! This indicates that our first found quadruple $(m, n, i, j)$ was already the answer. Could we have seen that back then?