Modular Arithmetic 4.1

The Modular Arithmetic Calculator offers a streamlined approach to performing arithmetic operations modulo N. By allowing users to select a fixed modulus, it alleviates the need to repeatedly engage a "mod" button during calculations. This calculator is characterized by several key features:

• Adherence to conventional order of operations;
• Support for arbitrarily large integers;
• Efficient execution of modular division and exponentiation;
• Capability to display a complete transcript of calculations.

Modular arithmetic, fundamentally understood as a "calculus of remainders," is integral across various fields of mathematics and computer science, with notable applications in cryptography, coding systems for barcodes, and even in music theory.

The core principle involves selecting a modulus N and thereby reducing each number to one of the integers in the range 0 to N−1 based on the remainder obtained from division by N. For illustrative purposes, consider a modulus of 17:

40 ≡ 6 (since dividing 40 by 17 yields a remainder of 6);

17 ≡ 0 (since dividing 17 by 17 results in no remainder).

The calculator respects these modular rules in arithmetic operations as well. Continuing with the modulus of 17:

15 + 7 ≡ 5 (since 22 reduces to 5);

3 × 9 ≡ 10 (as 27 reduces to 10);

5 ^ 3 ≡ 6 (because 125 reduces to 6).

Subtraction and division are also consistent with this modular framework:

−1 ≡ 16 (as 16 + 1 = 17 ≡ 0);

1/2 ≡ 9 (since 9 × 2 = 18 ≡ 1);

4 - 7 ≡ 14 (since 14 + 7 = 21 ≡ 4);

7 ÷ 3 = 8 (as 8 × 3 = 24 ≡ 7).

The concept excludes negative numbers and fractions; such cases are also reduced to one of the integers from the set {0,1,...,N−1}. Division by zero is prohibited, as is division when the divisor shares common factors with the modulus. When employing a modulus of 10, errors arise in the following operations:

• 3 ÷ 20 (as 20 ≡ 0);
• 7 ÷ 8 (because 8 and 10 share a common factor of 2).

The calculator accommodates integers of any size. For example, setting the modulus to a Mersenne prime, specifically 2305843009213693951, illustrates that:

5 ^ 2305843009213693950 ≡ 1, as per Fermat's little theorem.

The underlying code is meticulously designed and verified through an extensive suite of no fewer than 186 automated tests.

This application further enhances user experience by supporting external keyboards, Siri Shortcuts, as well as multitasking features on iPad such as Slide Over, Split View, and multiple windows.