Number Theory

 Number Theory Solver

Number Theory

121. Prime Factorization: $n=p_1^{k_1}{p}_2^{k_2}\dots p_m^{k_m}$

122. Greatest Common Divisor (GCD): $\gcd (a,b)$

123. Least Common Multiple (LCM): $\text{lcm}(a,b)$

124. Euclidean Algorithm: $\gcd (a,b)=\gcd (b,a\mathrm{m}\mathrm{o}\mathrm{d}b)$

125. Modular Arithmetic: $a\equiv b(\mathrm{m}\mathrm{o}\mathrm{d}m)$

126. Fermat's Little Theorem: $a^{p-1}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$

127. Euler's Totient Function: $\varphi (n)$

128. Chinese Remainder Theorem: $x\equiv a_i(\mathrm{m}\mathrm{o}\mathrm{d}{m}_i)$

129. Wilson's Theorem: $(p-1)!\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}p)$

130. Divisor Function: $\sigma (n)={\sum }_{d\mathrm{\mid }n}d$

131. Möbius Function: $\mu (n)$

132. Riemann Zeta Function: $\zeta (s)={\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$

133. Goldbach's Conjecture: Every even integer > 2 is the sum of two primes.

134. Twin Prime Conjecture: Infinitely many primes $p$ such that $p+2$ is also prime.

135. Perfect Number: $n={\sum }_{d\mathrm{\mid }n,d<n}d$

136. Abundant Number: ${\sum }_{d\mathrm{\mid }n,d<n}d>n$

137. Deficient Number: ${\sum }_{d\mathrm{\mid }n,d<n}d<n$

138. Pythagorean Triples: $a^2+b^2=c^2$

139. Fibonacci Sequence: $F_n=F_{n-1}+F_{n-2}$

140. Lucas Sequence: $L_n=L_{n-1}+L_{n-2}$