Problem 141: Investigating Progressive Numbers

Mathematical Development

Definition 1. We call $n$ progressive if there exist integers $q, d, r$ with $n = qd + r$, $0 \leq r < d$, $q > d > r \geq 0$, and $d^2 = qr$ (the geometric sequence condition).

Theorem 1 (Exclusion of $r = 0$). No progressive number has $r = 0$.

Proof. If $r = 0$, then $d^2 = qr = 0$, so $d = 0$. But $d > r \geq 0$ requires $d \geq 1$, a contradiction. $\qed$

Theorem 2 (Parametrization). Every progressive number $n$ with $r \geq 1$ admits a unique representation

where $a, b, c$ are positive integers satisfying $a > b$, $\gcd(a, b) = 1$, and $c \geq 1$. The corresponding division parameters are $q = a^2 c$, $d = abc$, $r = b^2 c$.

Proof. Suppose $q > d > r \geq 1$ with $d^2 = qr$ and $n = qd + r$.

Step 1 (Reduction to coprime form). Let $g = \gcd(d, r)$ and write $d = g\alpha$, $r = g\beta$ where $\gcd(\alpha, \beta) = 1$. From $d^2 = qr$:

Since $\gcd(\alpha, \beta) = 1$ and $q$ is a positive integer, we require $\beta \mid g$. Write $g = \beta c$ for some positive integer $c$.

Step 2 (Explicit formulas). Substituting $g = \beta c$:

Step 3 (Relabeling). Setting $a = \alpha$ and $b = \beta$ yields $q = a^2 c$, $d = abc$, $r = b^2 c$ with $\gcd(a,b) = 1$.

Step 4 (Ordering verification). We verify $q > d > r$:

• $q > d \iff a^2 c > abc \iff a > b$, which holds by construction since $q > d$ in the original problem.
• $d > r \iff abc > b^2 c \iff a > b$, also satisfied.

Step 5 (Formula for $n$).

Step 6 (Uniqueness). Given $(q, d, r)$, the ratio $a/b = d/r$ in lowest terms is uniquely determined, and $c = r/b^2$ follows. Conversely, distinct $(a,b,c)$ triples yield distinct $(q,d,r)$ triples since $q = a^2 c$, $d = abc$, $r = b^2 c$ are injective in $(a,b,c)$. $\qed$

Lemma 1 (Search bounds). The following bounds hold:

1. $a \leq \lfloor N^{1/3} \rfloor$ where $N = 10^{12}$.
2. For fixed $(a,b)$: $c \leq \lfloor \sqrt{N / (a^3 b)} \rfloor$.

Proof. Since $n = a^3 bc^2 + b^2 c \geq a^3 bc^2 \geq a^3 \cdot 1 \cdot 1 = a^3$, the constraint $n < N$ forces $a < N^{1/3}$. For fixed $(a,b)$ with $c \geq 1$, $n \geq a^3 bc^2$ gives $c \leq \sqrt{N/(a^3 b)}$. $\qed$

Remark. Different $(a,b,c)$ triples may produce the same value of $n$ (though different $(q,d,r)$ triples). Hence we collect progressive numbers in a set to avoid double-counting before checking for perfect squares.

Editorial

n is progressive if n = qd + r where q, d, r form a geometric sequence with q > d > r >= 0. Parametrization: n = a^3bc^2 + b^2c with gcd(a,b)=1, a>b>=1, c>=1. We enumerate the admissible parameter triples, test each generated value for being a square, and sum the distinct progressive squares that satisfy the bound.

Pseudocode

INPUT:  N = 10^12
OUTPUT: Sum of all progressive perfect squares below N
S <- empty set
FOR a = 2, 3, ..., floor(N^{1/3}):
FOR b = 1, 2, ..., a - 1:
IF gcd(a, b) != 1 THEN CONTINUE
A <- a^3 * b
IF A >= N THEN BREAK
FOR c = 1, 2, ...:
n <- A * c^2 + b^2 * c
IF n >= N THEN BREAK
s <- floor(sqrt(n))
IF s^2 = n THEN S <- S union {n}
RETURN sum of all elements in S

Complexity Analysis

Time. The outer loop over $a$ runs $O(N^{1/3})$ iterations. For each $a$, the loop over $b$ runs at most $a$ iterations, reduced by the coprimality filter to $O(a \cdot 6/\pi^2)$ on average. The innermost loop over $c$ runs $O(\sqrt{N/(a^3 b)})$ iterations. The total work is bounded by:

The inner sum is $O(\sqrt{a})$ and the resulting series converges, giving $W = O(\sqrt{N})$.

Space. $O(K)$ where $K$ is the number of progressive perfect squares found (empirically very small).

Answer

#include <bits/stdc++.h>
using namespace std;

int main() {
    // Find sum of all progressive perfect squares below 10^12
    // n = a^3 * b * c^2 + b^2 * c, with gcd(a,b)=1, a>b>=1, c>=1
    // Check if n is a perfect square

    long long LIMIT = 1000000000000LL; // 10^12
    set<long long> squares;

    for (long long a = 2; a * a * a < LIMIT; a++) {
        for (long long b = 1; b < a; b++) {
            if (__gcd(a, b) != 1) continue;
            // n = a^3*b*c^2 + b^2*c
            // For c=1: n = a^3*b + b^2
            long long a3b = a * a * a * b;
            if (a3b >= LIMIT) break;
            for (long long c = 1; ; c++) {
                long long n = a3b * c * c + b * b * c;
                if (n >= LIMIT) break;
                long long s = (long long)sqrt((double)n);
                // Check around s due to floating point
                for (long long t = max(1LL, s - 2); t <= s + 2; t++) {
                    if (t * t == n) {
                        squares.insert(n);
                        break;
                    }
                }
            }
        }
    }

    long long ans = 0;
    for (long long x : squares) ans += x;
    cout << ans << endl;
    return 0;
}

"""
Problem 141: Investigating Progressive Numbers
Find the sum of all progressive perfect squares below 10^12.

n is progressive if n = q*d + r where q, d, r form a geometric sequence
with q > d > r >= 0.

Parametrization: n = a^3*b*c^2 + b^2*c with gcd(a,b)=1, a>b>=1, c>=1.
"""

from math import gcd, isqrt

def solve():
    LIMIT = 10**12
    progressive_squares = set()

    a = 2
    while a * a * a < LIMIT:
        for b in range(1, a):
            if gcd(a, b) != 1:
                continue
            a3b = a**3 * b
            if a3b >= LIMIT:
                break
            c = 1
            while True:
                n = a3b * c * c + b * b * c
                if n >= LIMIT:
                    break
                s = isqrt(n)
                if s * s == n:
                    progressive_squares.add(n)
                c += 1
        a += 1

    answer = sum(progressive_squares)
    print(answer)