Difference between Two Squares equal to Repunit/Examples
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Examples of Difference between Two Squares equal to Repunit
Repunit $R_1$
| \(\ds 1\) | \(=\) | \(\ds 1 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1 + 1} 2\) | \(=\) | \(\ds 1\) | |||||||||||
| \(\ds \frac {1 - 1} 2\) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1^2 - 0^2\) | \(=\) | \(\ds 1\) |
$\blacksquare$
Repunit $R_2$
We have that $11$ is a prime.
| \(\ds 11\) | \(=\) | \(\ds 1 \times 11\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {11 + 1} 2\) | \(=\) | \(\ds 6\) | |||||||||||
| \(\ds \frac {11 - 1} 2\) | \(=\) | \(\ds 5\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 6^2 - 5^2\) | \(=\) | \(\ds 36 - 25\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 11\) |
$\blacksquare$
Repunit $R_3$
We have that:
| \(\ds 111\) | \(=\) | \(\ds 3 \times 37\) | ||||||||||||
| \(\ds 111\) | \(=\) | \(\ds 1 \times 111\) |
| \(\ds 111\) | \(=\) | \(\ds 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {111 + 1} 2\) | \(=\) | \(\ds 56\) | |||||||||||
| \(\ds \frac {111 - 1} 2\) | \(=\) | \(\ds 55\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 56^2 - 55^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3136 - 3025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111\) |
| \(\ds 111\) | \(=\) | \(\ds 37 \times 3\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {37 + 3} 2\) | \(=\) | \(\ds 20\) | |||||||||||
| \(\ds \frac {37 - 3} 2\) | \(=\) | \(\ds 17\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 20^2 - 17^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 400 - 289\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111\) |
$\blacksquare$
Repunit $R_4$
We have that:
| \(\ds 1111\) | \(=\) | \(\ds 101 \times 11\) | ||||||||||||
| \(\ds 1111\) | \(=\) | \(\ds 1 \times 1111\) |
| \(\ds 1111\) | \(=\) | \(\ds 1111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1111 + 1} 2\) | \(=\) | \(\ds 556\) | |||||||||||
| \(\ds \frac {1111 - 1} 2\) | \(=\) | \(\ds 555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 556^2 - 555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 309 \, 136 - 308 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1111\) |
| \(\ds 1111\) | \(=\) | \(\ds 101 \times 11\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {101 + 11} 2\) | \(=\) | \(\ds 56\) | |||||||||||
| \(\ds \frac {101 - 11} 2\) | \(=\) | \(\ds 45\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 56^2 - 45^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3136 - 2025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1111\) |
$\blacksquare$
Repunit $R_5$
We have that:
| \(\ds 11 \, 111\) | \(=\) | \(\ds 41 \times 271\) | ||||||||||||
| \(\ds 11 \, 111\) | \(=\) | \(\ds 1 \times 11 \, 111\) |
| \(\ds 11 \, 111\) | \(=\) | \(\ds 11 \, 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {11 \, 111 + 1} 2\) | \(=\) | \(\ds 5556\) | |||||||||||
| \(\ds \frac {11 \, 111 - 1} 2\) | \(=\) | \(\ds 5555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 5556^2 - 5555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 30 \, 869 \, 136 - 30 \, 858 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \, 111\) |
| \(\ds 11 \, 111\) | \(=\) | \(\ds 271 \times 41\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {271 + 41} 2\) | \(=\) | \(\ds 156\) | |||||||||||
| \(\ds \frac {271 - 41} 2\) | \(=\) | \(\ds 115\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 156^2 - 115^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 24 \, 336 - 13 \, 225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \, 111\) |
$\blacksquare$
Repunit $R_6$
We have that:
- $111 \, 111 = 3 \times 7 \times 11 \times 13 \times 37$
So, from $\sigma_0$ of $111 \, 111$, there are $32$ divisors of $111 \, 111$, which can be grouped in $16$ pairs.
Each of these will generate a Difference between Two Squares equal to $111 \, 111$.
Hence:
| \(\ds 111 \, 111\) | \(=\) | \(\ds 1 \times 111 \, 111\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \times 37 \, 037\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 7 \times 15 \, 873\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 11 \times 10 \, 101\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 13 \times 8547\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 21 \times 5291\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 33 \times 3367\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 37 \times 3003\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 39 \times 2849\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 77 \times 1443\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 91 \times 1221\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \times 1001\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 143 \times 777\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 231 \times 481\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 259 \times 429\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 273 \times 407\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 111 \, 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {111 \, 111 + 1} 2\) | \(=\) | \(\ds 55 \, 556\) | |||||||||||
| \(\ds \frac {111 \, 111 - 1} 2\) | \(=\) | \(\ds 55 \, 555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 55 \, 556^2 - 55 \, 555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 3 \, 086 \, 469 \, 136 - 3 \, 086 \, 358 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 37 \, 037 \times 3\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {37 \, 037 + 3} 2\) | \(=\) | \(\ds 18 \, 520\) | |||||||||||
| \(\ds \frac {37 \, 037 - 3} 2\) | \(=\) | \(\ds 18 \, 517\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 18 \, 520^2 - 18 \, 517^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 342 \, 990 \, 400 - 342 \, 879 \, 289\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 15 \, 873 \times 7\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {15 \, 873 + 7} 2\) | \(=\) | \(\ds 7940\) | |||||||||||
| \(\ds \frac {15 \, 873 - 7} 2\) | \(=\) | \(\ds 7933\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 7940^2 - 7933^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 63 \, 043 \, 600 - 62 \, 932 \, 489\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 10 \, 101 \times 11\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {10 \, 101 + 11} 2\) | \(=\) | \(\ds 5056\) | |||||||||||
| \(\ds \frac {10 \, 101 - 11} 2\) | \(=\) | \(\ds 5045\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 5056^2 - 5045^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 25 \, 563 \, 136 - 25 \, 452 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 8547 \times 13\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {8547 + 13} 2\) | \(=\) | \(\ds 4280\) | |||||||||||
| \(\ds \frac {8547 - 13} 2\) | \(=\) | \(\ds 4267\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 4280^2 - 4267^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 18 \, 318 \, 400 - 18 \, 207 \, 289\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 5291 \times 21\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {5291 + 21} 2\) | \(=\) | \(\ds 2656\) | |||||||||||
| \(\ds \frac {5291 - 21} 2\) | \(=\) | \(\ds 2635\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 2656^2 - 2635^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 7 \, 054 \, 336 - 6 \, 943 \, 225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 3367 \times 33\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {3367 + 33} 2\) | \(=\) | \(\ds 1700\) | |||||||||||
| \(\ds \frac {3367 - 33} 2\) | \(=\) | \(\ds 1667\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 1700^2 - 1667^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \, 890 \, 000 - 2 \, 778 \, 889\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 3003 \times 37\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {3003 + 37} 2\) | \(=\) | \(\ds 1520\) | |||||||||||
| \(\ds \frac {3003 - 37} 2\) | \(=\) | \(\ds 1483\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 1520^2 - 1483^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \, 310 \, 400 - 2 \, 199 \, 289\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 2849 \times 39\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {2849 + 39} 2\) | \(=\) | \(\ds 1444\) | |||||||||||
| \(\ds \frac {2849 - 39} 2\) | \(=\) | \(\ds 1405\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 1444^2 - 1405^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \, 085 \, 136 - 1 \, 974 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 1443 \times 77\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1443 + 77} 2\) | \(=\) | \(\ds 760\) | |||||||||||
| \(\ds \frac {1443 - 77} 2\) | \(=\) | \(\ds 683\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 760^2 - 683^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 577 \, 600 - 466 \, 489\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 1221 \times 91\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1221 + 91} 2\) | \(=\) | \(\ds 656\) | |||||||||||
| \(\ds \frac {1221 - 91} 2\) | \(=\) | \(\ds 565\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 656^2 - 565^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 430 \, 336 - 319 \, 225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 1001 \times 111\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1001 + 111} 2\) | \(=\) | \(\ds 556\) | |||||||||||
| \(\ds \frac {1001 - 111} 2\) | \(=\) | \(\ds 445\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 556^2 - 445^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 309 \, 136 - 198 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 777 \times 143\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {777 + 143} 2\) | \(=\) | \(\ds 460\) | |||||||||||
| \(\ds \frac {777 - 143} 2\) | \(=\) | \(\ds 445\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 460^2 - 317^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 211 \, 600 - 100 \, 489\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 481 \times 231\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {481 + 231} 2\) | \(=\) | \(\ds 356\) | |||||||||||
| \(\ds \frac {481 - 231} 2\) | \(=\) | \(\ds 125\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 356^2 - 125^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 126 \, 736 - 15 \, 625\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 429 \times 259\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {429 + 259} 2\) | \(=\) | \(\ds 344\) | |||||||||||
| \(\ds \frac {429 - 259} 2\) | \(=\) | \(\ds 85\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 344^2 - 85^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 118 \, 336 - 7225\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
| \(\ds 111 \, 111\) | \(=\) | \(\ds 407 \times 273\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {407 + 273} 2\) | \(=\) | \(\ds 340\) | |||||||||||
| \(\ds \frac {407 - 273} 2\) | \(=\) | \(\ds 67\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 340^2 - 67^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 115 \, 600 - 4489\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 111 \, 111\) |
$\blacksquare$
Repunit $R_7$
We have that:
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 239 \times 4649\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times 1 \, 111 \, 111\) |
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 1 \, 111 \, 111 \times 1\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {1 \, 111 \, 111 + 1} 2\) | \(=\) | \(\ds 555 \, 556\) | |||||||||||
| \(\ds \frac {1 \, 111 \, 111 - 1} 2\) | \(=\) | \(\ds 555 \, 555\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 555 \, 556^2 - 555 \, 555^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 308 \, 642 \, 469 \, 136 - 308 \, 641 \, 358 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \, 111 \, 111\) |
| \(\ds 1 \, 111 \, 111\) | \(=\) | \(\ds 4649 \times 239\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac {4649 + 239} 2\) | \(=\) | \(\ds 2444\) | |||||||||||
| \(\ds \frac {4649 - 239} 2\) | \(=\) | \(\ds 2205\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds 2444^2 - 2205^2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \, 973 \, 136 - 4 \, 862 \, 025\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \, 111 \, 111\) |
$\blacksquare$