Open Problems in Number Theory

The Nature of Open Problems

Number theory contains some of the oldest and deepest unsolved problems in mathematics.

Many are deceptively simple to state. Elementary questions about divisibility, primes, rational points, or polynomial equations often resist centuries of effort.

Open problems serve several purposes:

• they organize research directions,
• they reveal structural gaps in current theory,
• they connect different branches of mathematics,
• they motivate new techniques and conjectures.

Modern number theory is shaped as much by its unsolved problems as by its established theorems.

The Riemann Hypothesis

The most famous open problem in number theory is the Riemann Hypothesis.

The Riemann zeta function is

Its nontrivial zeros lie in the critical strip

The hypothesis states that every nontrivial zero satisfies

The Riemann Hypothesis governs the distribution of prime numbers.

If true, it would imply extremely precise estimates for prime-counting functions and many other arithmetic quantities.

The hypothesis also generalizes to broad families of $L$-functions.

Twin Prime Conjecture

The twin prime conjecture asserts that infinitely many primes differ by $2$:

Examples include:

Heuristics strongly suggest infinitude, but no proof is known.

Major progress occurred when entity[“people”,“Yitang Zhang”,“Chinese mathematician”] proved the existence of infinitely many prime pairs separated by a bounded gap.

Subsequent work dramatically reduced the bound.

Nevertheless, the exact gap $2$ remains open.

Goldbach Conjecture

The Goldbach conjecture states:

Every even integer greater than $2$ is the sum of two primes.

Examples:

This problem has been tested computationally to enormous bounds and is strongly supported by probabilistic heuristics.

Partial results exist, but a complete proof remains unknown.

The weak Goldbach conjecture, concerning sums of three odd primes, was proved by entity[“people”,“Harald Helfgott”,“Peruvian mathematician”].

Prime Gaps

The behavior of gaps between consecutive primes remains mysterious.

Questions include:

• Are there infinitely many bounded gaps?
• How large can prime gaps become?
• How small can normalized gaps become?

Cramér conjectured that maximal prime gaps satisfy roughly

On the other hand, arbitrarily large gaps are known to exist.

Understanding the fluctuation of prime gaps is one of the major goals of analytic number theory.

Distribution of Primes

Many problems concern the fine distribution of primes.

Examples include:

• primes in short intervals,
• primes in arithmetic progressions,
• primes represented by polynomials,
• patterns among primes.

The Hardy-Littlewood conjectures provide detailed heuristics, but most remain unproved.

Even proving infinitely many primes of the form

is currently beyond known methods.

The Birch and Swinnerton-Dyer Conjecture

Let $E$ be an elliptic curve over $\mathbb{Q}$.

Its associated $L$-function is

The Birch and Swinnerton-Dyer conjecture states that the order of vanishing of

at

equals the rank of the elliptic curve.

Thus analytic behavior of the $L$-function predicts the number of rational points.

This conjecture is one of the Clay Millennium Problems and lies at the center of arithmetic geometry.

Hodge and Tate-Type Problems

Arithmetic geometry contains major open conjectures relating algebraic cycles, cohomology, and arithmetic structure.

Examples include:

• the Hodge conjecture,
• the Tate conjecture,
• the Fontaine-Mazur conjecture.

These problems connect algebraic geometry, Galois theory, and $L$-functions.

They are among the deepest structural questions in modern mathematics.

The ABC Conjecture

The ABC conjecture concerns integers satisfying

Define the radical

as the product of distinct prime factors of $abc$.

The conjecture predicts that usually

cannot be much larger than the radical.

Despite its elementary appearance, the conjecture has profound implications throughout number theory.

Consequences include results about:

• Diophantine equations,
• perfect powers,
• elliptic curves,
• Fermat-type equations.

Claims of proof remain controversial and are not universally accepted by the mathematical community.

Perfect Numbers and Mersenne Primes

A perfect number equals the sum of its proper divisors.

Euclid showed that even perfect numbers arise from Mersenne primes:

Open questions include:

• Are there infinitely many Mersenne primes?
• Do odd perfect numbers exist?

No odd perfect number has ever been found.

Rational Points

Diophantine geometry studies rational solutions to polynomial equations.

Given a variety $X$, one asks:

What is the structure of

Major open problems include:

• existence of rational points,
• distribution of rational points,
• effective determination of solutions,
• local-global principles.

Even when finiteness is known theoretically, explicit computation may remain difficult.

Effective Results

Many theorems guarantee existence or finiteness without providing efficient algorithms.

Open questions often concern effectiveness.

Examples include:

• effective Mordell conjecture,
• explicit bounds for rational points,
• effective versions of Chebotarev density,
• explicit class number bounds.

Computational number theory increasingly seeks quantitative versions of qualitative theorems.

Complexity-Theoretic Problems

Number theory intersects computational complexity in many unresolved ways.

Examples include:

Post-quantum cryptography depends heavily on understanding these hardness assumptions.

Randomness in Arithmetic

Many arithmetic phenomena appear random, but their true structure remains unclear.

Examples include:

• Möbius randomness,
• cancellation in character sums,
• pseudorandomness of primes,
• zero statistics of $L$-functions.

Understanding arithmetic randomness is a major challenge connecting number theory, probability, and ergodic theory.

Langlands Program

The Langlands program seeks a vast unification of number theory and representation theory.

It predicts deep correspondences between:

• automorphic forms,
• Galois representations,
• harmonic analysis,
• arithmetic geometry.

Many cases are known, but the general theory remains incomplete.

The Langlands program is often viewed as one of the grand organizing principles of modern mathematics.

Zeta Functions and $L$-Functions

Generalized Riemann hypotheses for broader classes of $L$-functions remain open.

Questions concern:

• zero distributions,
• special values,
• functional equations,
• arithmetic interpretations.

Modern arithmetic geometry increasingly interprets arithmetic information through associated $L$-functions.

Computational Frontiers

Large computations continue to drive discovery.

Open computational questions include:

• finding larger primes,
• factoring larger integers,
• computing high zeta zeros,
• constructing explicit arithmetic databases,
• testing conjectures experimentally.

Experimental mathematics increasingly complements theoretical work.

Why Open Problems Matter

Open problems shape mathematical progress.

Attempts to solve them often produce entirely new theories.

Examples include:

Even failed attempts frequently generate important mathematics.

Conceptual Importance

Open problems in number theory reveal the extraordinary depth hidden inside elementary arithmetic.

Questions about primes, divisibility, and polynomial equations connect to:

• complex analysis,
• algebraic geometry,
• topology,
• probability,
• representation theory,
• computation,
• physics.

Number theory therefore remains simultaneously ancient and modern: many of its central problems can be explained to beginners, yet their solutions require some of the deepest ideas in mathematics.