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Session Length
~17 min
Adaptive Checks
15 questions
Transfer Probes
8
Number theory is the branch of pure mathematics devoted to the study of the integers and integer-valued functions. Often called the 'queen of mathematics,' it investigates properties of numbers such as divisibility, the distribution of prime numbers, and the solvability of equations in whole numbers. From the ancient investigations of Euclid, who proved that there are infinitely many primes, to the modern breakthroughs surrounding the proof of Fermat's Last Theorem by Andrew Wiles, number theory has been a driving force in mathematical discovery for over two millennia.
The subject divides into several major branches. Elementary number theory studies divisibility, congruences, and arithmetic functions without relying on techniques from other mathematical areas. Analytic number theory employs tools from calculus and complex analysis to understand the distribution of primes, most famously through the Riemann zeta function and the Prime Number Theorem. Algebraic number theory extends the notion of integers to algebraic number fields, using structures such as rings of integers and ideals to attack problems like unique factorization. Computational number theory focuses on algorithms for primality testing, integer factorization, and discrete logarithms, which form the backbone of modern cryptographic systems.
Beyond its theoretical elegance, number theory has become essential to the modern digital world. The RSA cryptosystem, elliptic-curve cryptography, and the Diffie-Hellman key exchange all rest on number-theoretic hardness assumptions. Error-correcting codes, hash functions, and random-number generators also draw heavily on results from the field. At the same time, longstanding open problems such as the Riemann Hypothesis, the Goldbach Conjecture, and the Twin Prime Conjecture continue to inspire new research, ensuring that number theory remains one of the most active and rewarding areas of mathematical inquiry.
One step at a time.
Natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Primes are the fundamental building blocks of all integers through the Fundamental Theorem of Arithmetic.
Example: The number 29 is prime because no integer between 2 and 28 divides it evenly, whereas $28 = 2 \times 2 \times 7$ is composite.
Every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. This guarantees a single canonical prime factorization for every positive integer.
Example: The integer 360 factors uniquely as $2^3 \times 3^2 \times 5$, and no other combination of primes produces 360.
A system of arithmetic for integers in which numbers 'wrap around' upon reaching a certain value called the modulus. Two integers are congruent modulo $n$ if their difference is divisible by $n$.
Example: 17 is congruent to 2 modulo 5 (written $17 \equiv 2 \pmod{5}$) because $17 - 2 = 15$ is divisible by 5. Clock arithmetic is a familiar base-12 example.
An efficient method for computing the greatest common divisor (GCD) of two integers by repeatedly applying the division algorithm. It is one of the oldest known algorithms still in widespread use.
Example: To find $\gcd(252, 105)$: $252 = 2 \times 105 + 42$, then $105 = 2 \times 42 + 21$, then $42 = 2 \times 21 + 0$, so $\gcd = 21$.
If $p$ is a prime and $a$ is an integer not divisible by $p$, then $a^{p-1} \equiv 1 \pmod{p}$. This result underpins many primality tests and cryptographic protocols.
Example: With $a = 2$ and $p = 7$: $2^6 = 64$, and 64 divided by 7 leaves remainder 1, confirming $2^6 \equiv 1 \pmod{7}$.
The function $\varphi(n)$ counts the number of integers from 1 to $n$ that are coprime to $n$. It generalizes Fermat's Little Theorem and is central to the RSA cryptosystem.
Example: $\varphi(12) = 4$, because exactly four numbers less than 12 are coprime to it: 1, 5, 7, and 11.
Polynomial equations where only integer (or sometimes rational) solutions are sought. Named after Diophantus of Alexandria, they include famous problems such as Fermat's Last Theorem.
Example: The equation $x^2 + y^2 = z^2$ has infinitely many integer solutions called Pythagorean triples, such as $(3, 4, 5)$ and $(5, 12, 13)$.
A result stating that if one knows the remainders of an integer when divided by several pairwise coprime moduli, one can determine the remainder when divided by the product of those moduli uniquely.
Example: If $x \equiv 2 \pmod{3}$ and $x \equiv 3 \pmod{5}$, then $x \equiv 8 \pmod{15}$. The theorem guarantees a unique solution modulo 15.
More terms are available in the glossary.
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