### Gauss factorials and Motzkin numbers ### For Neil Sloane and Doron Zeilberger Certain Gauss factorials are related to Motzkin numbers The main point. For certain primes 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 (mod 4) (they are the primes in the OEIS's sequence A103772; the first five are 17, 241, 3361, 46817, 652081, and the next six have 150, 229, 339, 401, 594 and 806 decimal digits (these are all those with fewer than 1,000 decimal digits), and only for those primes, do the following Gauss factorial congruences hold (with the coefficients on the right-hand side being sequence A002026 of the OEIS Encyclopedia, the 'first diffecence' of the Motzkin numbers): 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1GLDYjUSFGJy8lJXNpemVHUSMxNEYnL0YzRjdGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) ... (i) [this a consequence of being 'Gauss level 4'] 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjJGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) ... (ii) [this (ii) played a vital role in the Hungarica paper mentioned below; it established that these primes are not 1-exceptional] 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjNGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjRGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUklbXN1cEdGJDYlLUYjNiotSSVtc3ViR0YkNiUtSShtZmVuY2VkR0YkNiYtRiM2Ky1JJm1mcmFjR0YkNigtSSNtbkdGJDYmUSIxRicvJSVzaXplR1EjMTRGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGRC1GPDYmUSI0RidGP0ZCRkUvJS5saW5ldGhpY2tuZXNzR0Y+LyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk4vJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYuUTEmSW52aXNpYmxlVGltZXM7RicvJSVib2xkR0ZTL0ZDUSdub3JtYWxGJy8lJmZlbmNlR0ZTLyUqc2VwYXJhdG9yR0ZTLyUpc3RyZXRjaHlHRlMvJSpzeW1tZXRyaWNHRlMvJShsYXJnZW9wR0ZTLyUubW92YWJsZWxpbWl0c0dGUy8lJ2FjY2VudEdGUy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZvLUYjNiktRjQ2KC1GIzYpLUYjNistRiM2KS1GLDYlLUkjbWlHRiQ2J1EicEYnRj8vJSdpdGFsaWNHUSV0cnVlRicvRkNRLGJvbGQtaXRhbGljRidGRS1GIzYnLUY8NiZRIjVGJ0Y/RkJGRUY/RmlwRlxxRkUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj8vJStmb3JlZ3JvdW5kR1EqWzAsMTI4LDBdRicvJSlyZWFkb25seUdGW3EvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGJ0ZCRkUtRlU2L1EoJm1pbnVzO0YnRj9GQkZFRmZuRmhuRmpuRlxvRl5vRmBvRmJvL0Zlb1EsMC4yMjIyMjIyZW1GJy9GaG9GYnJGO0Y/RmZxRmlxRltyRkJGRUY/RmZxRmlxRltyRkJGRUY/RkJGRS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGP0ZmcUZpcUZbckZCRkVGP0ZmcUZpcUZbckZCRkVGP0ZCRkUtRiM2KUZlcEY/RmZxRmlxRltyRkJGRS8lL3N1YnNjcmlwdHNoaWZ0R0ZlcS1GVTYvUSIhRidGP0ZCRkVGZm5GaG5Gam5GXG9GXm9GYG9GYm8vRmVvUSwwLjExMTExMTFlbUYnL0Zob0Zic0Y/RmZxRmlxRltyRkJGRS1GPDYmUSI4RidGP0ZCRkVGY3EtRlU2LlEiPUYnRkJGRUZmbkZobkZqbkZcb0Zeb0Zgb0Ziby9GZW9RLDAuMjc3Nzc3OGVtRicvRmhvRlt0LUZVNi5RIn5GJ0ZCRkVGZm5GaG5Gam5GXG9GXm9GYG9GYm9GZG9GZ28tRlU2LlEqJnVtaW51czA7RidGQkZFRmZuRmhuRmpuRlxvRl5vRmBvRmJvRmFyRmNyLUY8NiVGPkZCRkVGXnItRjw2JlEiMkYnRj9GQkZFRlRGZXBGXnJGYHFGVC1GLDYlRmVwRmV0RmNxRl5yLUY8NiZRIzEyRidGP0ZCRkVGVC1GLDYlRmVwLUY8NiZRIjNGJ0Y/RkJGRUZjcUZeci1GPDYmUSMzMEYnRj9GQkZFRlQtRiw2JUZlcEZHRmNxLUZmcDYjUSFGJ0Y/RmZxRmlxRltyRkJGRQ== (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjVGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjZGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) 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 (mod LUklbXN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiZRIjdGJy8lJXNpemVHUSMxNEYnL0YzRjdGNUY/RkJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRic=) etc where the coefficients (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 ) are the first differences of the Motzkin numbers. Those primes are the level 4 Gauss primes of the Acta Mathematica Hungarica paper (see its Tables 5 and 6) THE GAUSS\342\200\223WILSON THEOREM FOR QUARTER-INTERVALS, 142 (1) (2014), 199\342\200\223230, by Cosgrave-Dilcher (both Doron and Neil have this; indeed Neil found a 'typo'). A quick reminder. They are the primes which satisfy (i) above, and (by Theorem 3 and Corollary 1 of the Hungarica paper) they are the primes 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, 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, 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, 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 (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). _________________ I was reminder of this when I saw Doron's mention of Motzkin numbers in recent Opinion 147. Of course I know how to prove the above expansions, and I only mention here one small ingredient, this perhaps already known relation): Let {LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JJW1yb3dHRiQ2KS1GLDYmUSJuRidGL0YyRjUvJSVzaXplR1EjMTRGJy8lK2ZvcmVncm91bmRHUSpbMCwxMjgsMF1GJy8lKXJlYWRvbmx5R0YxLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RicvRjNGN0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JlEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JI21vR0YkNi5RLyZHcmVhdGVyRXF1YWw7RicvRjNGN0Y1LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYlUSIwRidGPEY1LyUlc2l6ZUdRIzE0RicvJStmb3JlZ3JvdW5kR1EqWzAsMTI4LDBdRicvJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjxGNQ==} be the Motzkin numbers, and let 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 {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} be the 'first difference', then 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=