# Giuga Number/Examples/244,197,000,982,499,715,087,866,346

## 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:

 $\ds 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 122 \, 098 \, 500 \, 491 \, 249 \, 857 \, 543 \, 933 \, 173$ $\ds 2 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 81 \, 399 \, 000 \, 327 \, 499 \, 905 \, 029 \, 288 \, 782$ $\ds 2 \times 3 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 22 \, 199 \, 727 \, 362 \, 045 \, 428 \, 644 \, 351 \, 486$ $\ds 2 \times 3 \times 11 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 10 \, 617 \, 260 \, 912 \, 282 \, 596 \, 308 \, 168 \, 102$ $\ds 2 \times 3 \times 11 \times 23 \times 47 \, 137 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 7 \, 877 \, 322 \, 612 \, 338 \, 700 \, 486 \, 705 \, 366$ $\ds 2 \times 3 \times 11 \times 23 \times 31 \times 28 \, 282 \, 147 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 5 \, 180 \, 580 \, 032 \, 299 \, 461 \, 465 \, 258$ $\ds 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 3 \, 892 \, 535 \, 183$ $=$ $\ds 8 \, 634 \, 316 \, 234 \, 283 \, 759 \, 118$ $\ds 2 \times 3 \times 11 \times 23 \times 31 \times 47 \, 137 \times 28 \, 282 \, 147$ $=$ $\ds 62 \, 734 \, 693 \, 330 \, 195 \, 062$

 $\ds$  $\ds 122 \, 098 \, 500 \, 491 \, 249 \, 857 \, 543 \, 933 \, 173$ $\ds$  $\, \ds + \,$ $\ds 81 \, 399 \, 000 \, 327 \, 499 \, 905 \, 029 \, 288 \, 782$ $\ds$  $\, \ds + \,$ $\ds 22 \, 199 \, 727 \, 362 \, 045 \, 428 \, 644 \, 351 \, 486$ $\ds$  $\, \ds + \,$ $\ds 10 \, 617 \, 260 \, 912 \, 282 \, 596 \, 308 \, 168 \, 102$ $\ds$  $\, \ds + \,$ $\ds 7 \, 877 \, 322 \, 612 \, 338 \, 700 \, 486 \, 705 \, 366$ $\ds$  $\, \ds + \,$ $\ds 5 \, 180 \, 580 \, 032 \, 299 \, 461 \, 465 \, 258$ $\ds$  $\, \ds + \,$ $\ds 8 \, 634 \, 316 \, 234 \, 283 \, 759 \, 118$ $\ds$  $\, \ds + \,$ $\ds 62 \, 734 \, 693 \, 330 \, 195 \, 062$ $\ds$ $=$ $\ds 244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 347$ $\ds$ $=$ $\ds 244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 346 + 1$
Dividing both sides by $244 \, 197 \, 000 \, 982 \, 499 \, 715 \, 087 \, 866 \, 347$ makes the result apparent by definition of Giuga number.
$\blacksquare$