The sum of the squares of two consecutive positive integers is 145. What are the integers?

Tue, 05 May 2026 07:30:49 +0000

**Why the Curve of 145 Sparks a Quiet Math Mystery Across America**
The sum of the squares of two consecutive positive integers is 145. What are the integers? It’s a simple question—but one that’s quietly fueling curiosity in homes, classrooms, and online forums across the United States. With rising interest in puzzles, number patterns, and early STEM learning, this equation has become a subtle trend in casual discovery searches. People aren’t just solving equations—they’re unpacking symmetry, patterns, and the beauty of structured math in everyday life. It’s a miner’s Einstein moment for numeric thinkers, revealing how ancient logic still shapes modern curiosity. **Why This Problem Is Gaining Momentum in U.S. Culture**
In a digital age saturated with rapid-fire content, this type of mental puzzle stands out as a calm, intentional challenge. The root of its popularity lies in the public’s fresh interest in logic and pattern recognition, amplified by educational shows, parenting blogs, and social platforms emphasizing critical thinking. The combination of two consecutive numbers (say, n and n+1) introduces intuitive complexity—enough to engage without intimidation. It fits seamlessly into broader conversations about algebra-ready problem-solving, backed by mobile-friendly search behavior. With keywords like “sum of squares of two consecutive numbers equaling 145” growing steadily, this topic holds real potential to claim #1 real estate in Discover for US audiences. **Understanding How the Equation Works: A Clear, Neutral Breakdown**
Let’s examine the math with clarity and simplicity. We’re searching for two consecutive positive integers, n and n+1, such that:
n² + (n+1)² = 145
Expanding:
n² + n² + 2n + 1 = 145
Which simplifies to:
2n² + 2n + 1 = 145
Subtracting 145:
2n² + 2n – 144 = 0
Divide by 2:
n² + n – 72 = 0
Now, factor the quadratic:
(n + 9)(n – 8) = 0
Solutions: n = –9 or n = 8
Since we want positive integers, n = 8.
Thus, the numbers are 8 and 9.
Checking: 8² + 9² = 64 + 81 = 145—confirmed. This elegant solution reveals how number sequences create measurable outcomes, satisfying both curiosity and logical precision. **Common Questions About the Sum of Squares: What People Really Want to Know**