Exercise 4. This software finds the idempotents in Zn[i] = {a+bi| a, b belong to Zn} (Gaussian integers modulo n ). Run the software for n = 4, 8, 16, and 32. Make a conjecture about the number of idempotents when n = 2k. Run the software for n = 13, 17, 29, and 37. What do these values of n have in common? Make a conjecture about the number of idempotents for these n. Run the software for n = 7, 11, 19, 23, 31, and 43. What do these values of n have in common? Make a conjecture about the number of idempotents for these n.

Please input values for n here:

Example: if you want to run this software for n = 4, 8, 16, and 32, please input 4, 8, 16, 32 into the text box above and press the "What's the result?" button.