Giuga Number/Examples/244,197,000,982,499,715,087,866,346
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Example of Giuga Number
$244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346$ is a Giuga number:
- $\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 {11} + \dfrac 1 {23} + \dfrac 1 {31} + \dfrac 1 {47 \, 137} + \dfrac 1 {28 \, 282 \, 147} + \dfrac 1 {3 \, 892 \, 535 \, 183} - \dfrac 1 {244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346} = 1$
Proof
We have that:
- $244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346 = 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$
Then we have:
| \(\ds 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 122 \, 098 \, 500 \, 491 \, 249 \, 857 \, 543 \, 933 \, 173\) | ||||||||||||
| \(\ds 2 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 81 \, 399 \, 000 \, 327 \, 499 \, 905 \, 029 \, 288 \, 782\) | ||||||||||||
| \(\ds 2 \times 3 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 22 \, 199 \, 727 \, 362 \, 045 \, 428 \, 644 \, 351 \, 486\) | ||||||||||||
| \(\ds 2 \times 3 \times 11 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 10 \, 617 \, 260 \, 912 \, 282 \, 596 \, 308 \, 168 \, 102\) | ||||||||||||
| \(\ds 2 \times 3 \times 11 \times 23 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 7 \, 877 \, 322 \, 612 \, 338 \, 700 \, 486 \, 705 \, 366\) | ||||||||||||
| \(\ds 2 \times 3 \times 11 \times 23 \times 31 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 5 \, 180 \, 580 \, 032 \, 299 \, 461 \, 465 \, 258\) | ||||||||||||
| \(\ds 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 3 \, 892 \, 535 \, 183\) | \(=\) | \(\ds 8 \, 634 \, 316 \, 234 \, 283 \, 759 \, 118\) | ||||||||||||
| \(\ds 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147\) | \(=\) | \(\ds 62 \, 734 \, 693 \, 330 \, 195 \, 062\) |
Adding all these up together:
| \(\ds \) | \(\) | \(\ds 122 \, 098 \, 500 \, 491 \, 249 \, 857 \, 543 \, 933 \, 173\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 81 \, 399 \, 000 \, 327 \, 499 \, 905 \, 029 \, 288 \, 782\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 22 \, 199 \, 727 \, 362 \, 045 \, 428 \, 644 \, 351 \, 486\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 10 \, 617 \, 260 \, 912 \, 282 \, 596 \, 308 \, 168 \, 102\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 7 \, 877 \, 322 \, 612 \, 338 \, 700 \, 486 \, 705 \, 366\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 5 \, 180 \, 580 \, 032 \, 299 \, 461 \, 465 \, 258\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 8 \, 634 \, 316 \, 234 \, 283 \, 759 \, 118\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 62 \, 734 \, 693 \, 330 \, 195 \, 062\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 347\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346 + 1\) |
Dividing both sides by $244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 347$ makes the result apparent by definition of Giuga number.
$\blacksquare$
Sources
- Jan. 1996: D. Borwein, J.M. Borwein, P.B. Borwein and R. Girgensohn: Giuga's Conjecture on Primality (Amer. Math. Monthly Vol. 103, no. 1: pp. 40 – 50) www.jstor.org/stable/2975213
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $244,197,000,982,499,715,087,866,346$