[Appeared in J. Difference Equations and Applications v. 20 (2014), 973-988.]

In addition to the great interest of the actual theorems, (isn't it FASCINATING that the Motzkin numbers are NEVER divisible by 8?, and AMAZING that no Catalan number can ever give remainder 22 when it is divided by 256?), this project is yet another CASE STUDY in future mathematics, where the human does not waste his time proving new theorems, but teaches his bag of tricks to his friend the computer, who can prove much deeper theorems.

Maple Packages

• AutoSquared, a Maple package to automatically generate automata that enables one to determine super-fast, the congruence properties, modulo powers of prime (and hence, thanks to the Chinese Remainder Theorem, modulo any integer of famous combinatorial sequences like the Catalan, Motzkin, Delannoy, etc.etc. sequences.

• CatalanCC, a Maple package that uses Congruence Automata to compute super-fast the remainder, upon dividing Catalan numbers by an integer m that is a divisor of
  2⁶ 3³ 5²(7)(11)...(59)=2768774904222066200260800 .

• CatalanLS, a Maple package that uses Congruence Linear Schemes to compute super-fast the remainder, upon dividing Catalan numbers by an integer m that is a divisor of
  2³3³5³7²= 1323000 .

• MotzkinCC, a Maple package that uses Congruence Automata to compute super-fast the remainder, upon dividing Motzkin numbers by an integer m that is a divisor of
  2³ 3³ (5)(7)(11)...(29)=232908956280.

• MotzkinLS, a Maple package that uses Congruence Linear Schemes to compute super-fast the remainder, upon dividing Motzkin numbers by an integer m that is a divisor of
  2³ 3³ 5³ 7² =1323000 .

• DelannoyLS, a Maple package that uses Congruence Linear Schemes to compute super-fast the remainder, upon dividing the Central Delannoy numbers by an integer m that is a divisor of
  2³ 3³ 5³ 7² =1323000 .

Sample Input and Output for the Maple package AutoSquared

• If you want to get a list of automata for the Catalan numbers, modulo 2,3,5, and 7, and their powers ≤ 3, where the max. size is 500, the
  the input gives the output.

• If you want to get a list of automata for the Catalan numbers, modulo 2,3,5,7, and 11, and their powers ≤ 4, where the max. size is 1000, the
  the input gives the output.

• If you want to get a list of Congruence Linear Schemes for the Catalan numbers, phrased in terms of the letter A, modulo 2,3,5, and 7, and their powers ≤ 4, where the max. size is 500, the
  the input gives the output.

• If you want to get a list of Congruence Linear Schemes for the Catalan numbers, phrased in terms of the letter A, modulo powers of 2 up to 2⁸
  the input gives the output.

• If you want to get a list of automata for the Motzkin numbers, modulo 2,3,5, and 7, and their power and their powers ≤ 3 , where the max. size is 500, the where the max. size of the automata is 1000,
  the input gives the output.

• If you want to get a list of automata for the Motzkin numbers, modulo 2,3,5, 7, and 11, and their power and their powers ≤ 4 , where the max. size of the automata is 1000,
  the input gives the output.

• If you want to get a list of the Congruence Linear Schemes for the Motzkin numbers, modulo 2,3,5, and 7, and their power and their powers ≤ 3 , where the max. size of the automata is 1000,
  the input gives the output.

• If you want to get a list of the Congruence Linear Schemes for the Motzkin numbers, modulo p² for all primes from 2 to 29, owers ≤ 3 , where the max. size is 2000,
  the input gives the output.

• If you want to get a list of automata for the Central Delannoy numbers, modulo 2,3,5, and 7, and their power and their powers < 3, where the max. size is 500, the where the max. size of the automata is 1000,
  the input gives the output.

• If you want to get a list of Congruence Linear Schemes for the Central Delannoy numbers, modulo 2,3,5, and 7, and their power and their powers < 3, where the max. size is 500
  the input gives the output.

• If you want to get automata for many sequences modulo 2,4,8,16,3,9,27,5,25 given by constant terms
  the input gives the output.

• If you want to get Congruence Linear Schemes for many sequences modulo 2,4,8,16,3,9,27,5,25 given by constant terms
  the input gives the output.

• If you want to get a webbook with HUNDREDS of exciting theorems about automatic p-schemes for even more sequences modulo 2,4,8,16,3,9,27,5,25 given by constant terms
  the input gives the output.

• If you want to get THOUSANDS of exciting theorems about Congruence Linear Schemes, for many combinatorial sequences modulo 2,4,8,16,3,9,27,5,25 given by constant terms
  the input gives the output.

• If you want to get Congruence Automata for the Zeta-2 Apery numbers for the moduli 2,4,8,3,9, 5,25
  the input gives the output.

• If you want to get Congruence Linear Schemes for the Zeta-2 Apery numbers for the moduli 2,4,8,3,9, 5,25
  the input gives the output.

• If you want to get Congruence Automata for the (usual, Zeta(3)) Apery numbers for the moduli 2,4,8,3,9, 5,25
  the input gives the output.

• If you want to get Congruence Linear Schemes for the (usual, Zeta(3)) Apery numbers for the moduli 2,4,8,16,32
  the input gives the output.

• If you want to get Congruence Automata for the Catalan numbers for the moduli 2,4,8,16,32,64,128, and 256, and the avoided congruence classes
  the input gives the output.

• If you want to get Congruence Linear Schemes for the Catalan numbers for the moduli 2,4,8,16,32,64,128, and 256, and the avoided congruence classes
  the input gives the output.

• If you want to get Congruence Linear Schemes for the Motzkin numbers for the moduli 2,4,8,16,32,64, and 128, and the avoided congruence classes
  the input gives the output.

Sample Input and Output for the Maple package CatalanCC

• If you want to see the Catalan numbers from googol to googol plus a thousand modulo 221
  the input gives the output.
  (as you can see they are all zero!)

Sample Input and Output for the Maple package CatalanLS

• If you want to see the Catalan numbers from 5¹⁰⁰ to 5¹⁰⁰ plus a thousand modulo 1000 (i.e. the last three digits), the
  the input gives the output.
  

• If you want to see the 5¹⁰⁰ -th number mod 1000 (i.e. the last three digits), the
  the input gives the output.
  

Sample Input and Output for the Maple package MotzkinCC

• If you want to see the Motzkin numbers from googol to googol plus a thousand modulo 221
  the input gives the output.
  

Sample Input and Output for the Maple package MotzkinLS

• If you want to see the Motzkin numbers from 10³⁰ to 10³⁰+1000 modulo 1000 (i.e. the last three digits), the
  the input gives the output.
  

• If you want to see the googol-th Motzkin number mod 1000, (i.e. the last three digits) (it is 187) the
  the input gives the output.
  

Sample Input and Output for the Maple package DelannoyLS

• If you want to see the Central Delannoy numbers from 10³⁰ to 10³⁰ plus a thousand modulo 1000 (i.e. the last three digits), the
  the input gives the output.
  

• If you want to see the googol-th Central Delannoy number mod 1000 (i.e. the last three (decimal) digits) (it is 281), the
  the input gives the output.
  

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