level: olympiad

description: count of linear form values.

description_latex: |\{x \in [a,b] : \exists\, k,\; x = \alpha + k\beta\}|


level: standard

description: MinOverSet of sum with fixed product constraint.

description_latex: \min(a + b) \text{ s.t. } ab = N


description: MaxOverSet(SolutionsSet(n, And(n >= 2, n <= upper, IsPrime(n)))).

description_latex: \max\{p \le n : p \text{ prime}\}


description: CountOverSet({p : Exists(q, p*q=N ∧ gcd(p,q)=1 ∧ p<q)})


description: minimum prime factor of n via min divisor >= 2.

description_latex: \min\{p : p \mid n,\; p \text{ prime}\}


description: Summation(Mul(EulerPhi(Var(k)), Floor(Div(n, Var(k)))), k, 1, n)

description_latex: \sum_{k=1}^{n} \varphi(k)\lfloor n/k \rfloor = \tfrac{n(n+1)}{2}


description: MinOverSet/MaxOverSet of symmetric expression with symmetric constraint.

description_latex: \min/\max f(x,y) \text{ s.t. symmetric constraint}


description: CountOverSet({(i,j) ∈ [1,a]×[1,b] : i+j=t})


description: SumOverDivisors(n, var, EulerPhi(Var(var)))

description_latex: \sum_{d \mid n} \varphi(d) = n


description: CountOverSet(SolutionsSet(var, And(IsPositive, IsOdd, ..., Eq(Sum, S))))


level: elementary

description: CountOverSet of CartesianProduct.

description_latex: |A \times B| = |A| \cdot |B|


Arithmetic sum: Σ_{k=0}^{n} k = n(n+1)/2


description: CountOverSet({k : 0 ≤ k ≤ n ∧ C(n,k) odd})


type: spawner

description: ∑_{k=0}^{n} (-1)^k C(n,k) = 0 for n > 0 (inclusion-exclusion)


description: SumOverSet(SolutionsSet(x, Eq(polynomial, 0))) pattern.

description_latex: \sum \text{roots}(P) = -a_{n-1}/a_n \text{ (Vieta)}


description: CountOverSet(SolutionsSet(var, And(bounds..., Divides...)))


description: CountOverSet(SolutionsSet(n, And(n >= 1, n <= N, Divides(d, Fib(n))))).

description_latex: |\{1 \le n \le N : d \mid F_n\}|


description: CountOverSet(SolutionsSet(var, And(bounds..., Eq(GCD(var, m), 1))))


description: ∑_{d|gcd(a,b)} μ(d) = 0 when gcd(a,b) > 1


description: largest proper divisor of n (max d | n with d < n).

description_latex: \max\{d : d \mid n,\; d < n\}


description: CountOverSet(SolutionsSet(var, And(bounds, Congruent(var, Floor(Div(var, 2)), m))))

description_latex: |\{x \in [a,b] : x \equiv \lfloor x/2 \rfloor \pmod{m}\}|


description: CountOverSet(SolutionsSet(n, And(n >= 2, n <= upper, IsPrime(n)))).

description_latex: \pi(n) = |\{p \le n : p \text{ prime}\}|


count of lattice points x in [1,M]^n with Q(x) <= K for a PSD form Q.


description: CountOverSet(SolutionsSet(var, And(lower ≤ Pow(var,e) ≤ upper)))

description_latex: |\{x \in [a,b] : x = k^e \text{ for some } k\}|


description: CountOverSet(SolutionsSet(var, And(


description: CountOverSet(SolutionsSet(var, And(bounds, Eq(Mod(DigitSum(var, 2), 2), k))))


description: count of starting values with exact minimal period r under an iterated polynomial map mod M.

description_latex: |\{a \in [0,N] : P^r(a) \equiv a \pmod{M},\; P^k(a) \not\equiv a \text{ for } 1\le k<r\}|


description: CountOverSet({(i,j) ∈ [1,a]×[1,b] : gcd(i,j)=1}).


count of distinct values of a PSD quadratic form over a bounded integer lattice.


description: SumOverSet(SolutionsSet(n, And(n >= 1, n <= N, n mod d = 0))).

description_latex: \sum_{\substack{1 \le k \le n \\ d \mid k}} k


description: MaxKDivides(Mul(a, b, ...), p)


description: phi(1) = 1

description_latex: \varphi(1) = 1


description: phi(2) = 1

description_latex: \varphi(2) = 1


level: advanced

description: MaxKDivides(Factorial(n), p) where n is symbolic.

description_latex: v_p(n!) = \sum_{i=1}^{\infty} \lfloor n/p^i \rfloor


description: count of starting values with exact minimal period r under an iterated polynomial map modulo p^k (Hensel lift).

description_latex: |\{a \in [0,N] : P^r(a) \equiv a \pmod{p^k},\; P^j(a) \not\equiv a \text{ for } 1\le j<r\}|


Geometric sum: Σ_{k=0}^{n} r^k = (r^{n+1}-1)/(r-1)


count of lattice points x in [1,M]^n solving F(x) = R for a quartic form F.


count of lattice points x in [1,M]^n with Q(x) = R for a PSD quadratic form Q.


description: largest divisor of P not exceeding sqrt(P).

description_latex: \max\{d : d \mid P,\; d^2 \le P\}


count of lattice points x in [1,M]^n solving F(x) = R for a cubic form F.


minimum factor gap: min |x-y| with xy = P.


description: MaxKDivides(Sub(Pow(a, n), Pow(b, n)), p)

description_latex: v_p(a^n - b^n) \text{ (Lifting the Exponent)}


description: SumOverSet(MapOverSet(SolutionsSet((i,j), domain=A×B), f(i)))

description_latex: \sum_{(i,j) \in A\times B} f(i) = |B|\sum_{i \in A} f(i)


description: count of unordered solutions of a symmetric PSD form Q(x) = R under variable-permutation ordering.

description_latex: |\{x_1 \le \dots \le x_n : x^T A x = R\}|


description: ∑_{d|n} μ(d) = 0 for n > 1 (Möbius annihilation)


description: pattern for maximum valuation query.

description_latex: \max_{p^v \mid n} v


description: minimum of a PSD quadratic form Q(x) = x^T A x over x in [1,M]^n.

description_latex: \min_{x \in [1,M]^n} x^T A x


description: MaxKDivides(Binom(n, k), p)


description: MinOverSet({n : v_p(n!) >= k})


description: count of orbits of exact period r under iterated P mod p, filtered by a Legendre-symbol sum congruence.

description_latex: |\{a \in [0,N] : \text{period}_P(a) = r,\; \textstyle\sum_{i=0}^{r-1}\bigl(\frac{P^i(a)}{p}\bigr) \equiv 0 \pmod m\}|


description: MaxKDivides(Pow(a, k), p)

description_latex: v_p(a^k) = k \cdot v_p(a)


description: SumOverSet(MapOverSet(SolutionsSet((i,j), domain=A×B), Mul(f(i), g(j))))

description_latex: \sum_{(i,j) \in A\times B} f(i)g(j) = \bigl(\sum_{i \in A} f(i)\bigr)\bigl(\sum_{j \in B} g(j)\bigr)


minimum of a cubic form F(x) (sum of cubes of linear forms) over x in [1,M]^n.


description: product of two independent sums, each reducible via its own antilemma.

description_latex: \prod_i \sum_j a_{ij} \text{ expansion}


description: C(n, 0) - 1 = 0


description: C(n, n) - 1 = 0


description: SumOverSet(SolutionsSet(n, And(n >= lower, n <= upper, IsPrime(n)))).

description_latex: \sum_{\substack{p \le n \\ p \text{ prime}}} p


description: μ(n)² = 0 when n has a square factor

description_latex: \mu(n)^2 = [n \text{ is squarefree}]


description: 0! = 1


description: C(n, n) = 1

description_latex: \binom{n}{n} = 1


description: C(n, 0) = 1


description: τ(n) mod 2 = 0 when n is not a perfect square

description_latex: \tau(n) \bmod 2 = [n \text{ is a perfect square}]


description: totient summatory function: sum of phi(k) over k = 1..n.

description_latex: \sum_{k=1}^{n} \varphi(k)


compositions count: number of positive integer tuples (x1,...,xk) with sum S.


wilson

description: Wilson's theorem: (p-1)! is congruent to -1 modulo prime p.

description_latex: (p-1)! \equiv -1 \pmod{p}


description: MaxKDivides(Sum(Pow(a, n), Pow(b, n)), p) where n is odd.

description_latex: v_p(a^n + b^n) \text{ for odd } n, \; p \mid a+b


description: Liouville lambda evaluates to -1 when Omega(n) is odd.

description_latex: \lambda(n) = (-1)^{\Omega(n)} = -1 \text{ when } \Omega(n) \text{ odd}


description: a / a = 1


description: Liouville lambda evaluates to 1 when Omega(n) is even.

description_latex: \lambda(n) = (-1)^{\Omega(n)} = 1 \text{ when } \Omega(n) \text{ even}


minimum of a quartic form F(x) (sum of 4th powers of linear forms) over x in [1,M]^n.


description: a^0 = 1


count of lattice points (x,y) in [1,Nx] x [1,Ny] under a half-plane constraint ax+by <= C.


Big Omega vanishes at n = 1 (no prime factors with multiplicity).


description: Big Omega equals 1 at any prime p.

description_latex: \Omega(p) = 1


description: a²+b²+c² = ab+bc+ca forces a=b=c; combined with linear constraint determines unique value.


description: count of starting values a with exact minimal period r under iterated P mod p and Legendre-sum filter.


small omega vanishes at n = 1 (no distinct prime factors).


description: small omega equals 1 at any prime power p^k (single distinct prime).

description_latex: \omega(p) = 1


description: MaxKDivides(Sub(Pow(a, n), Pow(b, n)), 2)


count of integers x in [1,N] satisfying an absolute-value inequality |ax - b| <= c.


closed form for the general arithmetic-progression sum Sigma_{k=start}^{n} (a*k + b).


count of integers x in [1,N] satisfying a quadratic inequality a x^2 + b x + c <= 0 with a > 0.


description: a * 0 = 0

description_latex: a \cdot 0 = 0


description: a - a = 0


description: phi(p) - (p-1) = 0 for prime p

description_latex: \varphi(p) = p - 1


description: a % a = 0

description_latex: a \bmod a = 0


telescope

telescoping sum Sigma_{k=start}^{n} (f(k+1) - f(k)) collapses to f(n+1) - f(start).


description: count of starting values with preperiod t and exact cycle length s under iterated polynomial map mod M.

description_latex: |\{a \in [0,N] : P^{t+s}(a) \equiv P^t(a) \pmod{M},\; P^{t+k}(a) \not\equiv P^t(a) \text{ for } 1\le k<s\}|