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Until a couple of minutes before my edits, this page made it seem like it was a conjecture that there are no squares after 121 that are of the form x⁴ + x³ + x² + x + 1. This is actually easy to prove!

Case 1: Let x = 2k

Then, x⁴ + x³ + x² + x + 1 = 16k⁴ + 8k³ + 4k² + 2k + 1. For k > 0, this is between (4k+k)² and (4k+k+1)².

Case 2: Let x = 2k+1

Then, x⁴ + x³ + x² + x + 1 = 16k⁴ + 40k³ + 40k² + 20k + 5. For k > 0, this is between (4k²+k)² and (4k²+k+1)². For k > 1, this is between (4k²+5k+1)² and (4k²+5k+2)².

As a result, it suffices to check k = 0 and k = 1 and there are no solutions other than x = 0 and x = 3. 173.76.131.225 (talk) 21:32, 18 March 2021 (UTC)[reply]

I spot a mistake, but more importantly, this is original research. I'll revert your edits for now, but feel free to reinstate them if you can include a reliable source for this information. Gap9551 (talk) 07:43, 19 March 2021 (UTC)[reply]

please add my account 45.112.29.69 (talk) 00:27, 25 September 2023 (UTC)[reply]

For information on creating a Wikipedia account, please see Wikipedia:Account. Certes (talk) 12:37, 25 September 2023 (UTC)[reply]