Difference between Two Squares equal to Repdigit
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Theorem
Some differences of two squares that each make a repdigit number include:
\(\ds 6^2 - 5^2\) | \(=\) | \(\ds 11\) | ||||||||||||
\(\ds 56^2 - 45^2\) | \(=\) | \(\ds 1111\) | ||||||||||||
\(\ds 556^2 - 445^2\) | \(=\) | \(\ds 111 \, 111\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) |
\(\ds 7^2 - 4^2\) | \(=\) | \(\ds 33\) | ||||||||||||
\(\ds 67^2 - 34^2\) | \(=\) | \(\ds 3333\) | ||||||||||||
\(\ds 667^2 - 334^2\) | \(=\) | \(\ds 333 \, 333\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) |
\(\ds 8^2 - 3^2\) | \(=\) | \(\ds 55\) | ||||||||||||
\(\ds 78^2 - 23^2\) | \(=\) | \(\ds 5555\) | ||||||||||||
\(\ds 778^2 - 223^2\) | \(=\) | \(\ds 555 \, 555\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) |
\(\ds 9^2 - 2^2\) | \(=\) | \(\ds 77\) | ||||||||||||
\(\ds 89^2 - 12^2\) | \(=\) | \(\ds 7777\) | ||||||||||||
\(\ds 889^2 - 112^2\) | \(=\) | \(\ds 777 \, 777\) | ||||||||||||
\(\ds \) | \(:\) | \(\ds \) |
Proof
Let $a, b$ be integers with $1 \le b < a \le 8$ and $a + b = 9$.
Then:
\(\ds \paren {1 + \sum_{k \mathop = 0}^n a 10^k}^2 - \paren {1 + \sum_{k \mathop = 0}^n b 10^k}^2\) | \(=\) | \(\ds \paren {1 + \sum_{k \mathop = 0}^n a 10^k - 1 - \sum_{k \mathop = 0}^n b 10^k} \paren {1 + \sum_{k \mathop = 0}^n a 10^k + 1 + \sum_{k \mathop = 0}^n b 10^k}\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{k \mathop = 0}^n \paren {a - b} 10^k} \paren {2 + \sum_{k \mathop = 0}^n 9 \times 10^k}\) | $a + b = 9$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{k \mathop = 0}^n \paren {a - b} 10^k} \paren {1 + 10^{n + 1} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {a - b} 10^k + \sum_{k \mathop = 0}^n \paren {a - b} 10^{k + n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {a - b} 10^k + \sum_{k \mathop = n + 1}^{2 n + 1} \paren {a - b} 10^k\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{2 n + 1} \paren {a - b} 10^k\) |
which is a repdigit number.
The examples above are instances with $\tuple {a, b} = \tuple {5, 4}, \tuple {6, 3}, \tuple {7, 2}, \tuple {8, 1}$.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1111$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1111$