Solve prod_n=1^1003n^3/n^3+1=

$$\prod _ { n = 1 } ^ { 100 } \frac { 3 n ^ { 3 } } { n ^ { 3 } + 1 } =$$

Evaluate

$$\frac{1018937685598429054306368885307352572917594287092177304924986281988755393695254766422092122524658069651334196862674880003735537309749389364337785952880942685157239418331942070540737246517594091333816942592000000000000000000000000000000000000000000000000}{4800464732520701525317069794576355963019855696091283483447005390800470454752559850795692958261590330404119433369935857464890297565852390821061631485377548328267018030855696284369073241676880771561994105127}\approx 2.122581338 \cdot 10^{47}$$

Factor

$$\frac{2^{194}\times 3^{163}\times 5^{48}\times 11^{18}\times 17^{10}\times 23^{8}\times 29^{6}\times 41^{4}\times 47^{4}\times 53^{2}\times 59^{2}\times 71^{2}\times 83^{2}\times 89^{2}}{7\times 13^{2}\times 19\times 31\times 37\times 43\times 61\times 67\times 73\times 101\times 103^{2}\times 109^{2}\times 127\times 139^{2}\times 151\times 157\times 163\times 181\times 193\times 199\times 211\times 223\times 229\times 241\times 271\times 283\times 307\times 313\times 331\times 367\times 373\times 379\times 397\times 409\times 421\times 463\times 523\times 547\times 601\times 613\times 631\times 751\times 757\times 919\times 1069\times 1123\times 1249\times 1303\times 1483\times 1657\times 1723\times 1801\times 1951\times 2269\times 2437\times 2551\times 2791\times 2971\times 3169\times 3307\times 3541\times 3907\times 4423\times 4831\times 5113\times 5701\times 6007\times 6163\times 6481\times 8011\times 8191\times 9901}$$