Biography:Takashi Agoh

Takashi Agoh
吾郷 孝視
Born	1941 Izumo, Shimane Prefecture, Japan
Citizenship	Japanese
Alma mater	Tokyo University of Science (BS, PhD)
Known for	Agoh–Giuga conjecture Bernoulli numbers Euler polynomials
Title	Professor Emeritus
Scientific career
Fields	Number theory
Institutions	Tokyo University of Science

Takashi Agoh (born 1941) is a Japanese mathematician in the Department of Mathematics at the Tokyo University of Science in Noda, Chiba, Japan. His research focuses on the structural properties of Bernoulli numbers, Euler polynomials, and the distribution of prime numbers.

He is known for independently formulating a primality conjecture in 1990 that was later shown to be equivalent to a conjecture stated by Giuseppe Giuga in 1950; the combined statement is now known as the Agoh–Giuga conjecture. He has also published extensively on convolution identities, recurrence relations for special sequences, and the arithmetic of Fermat and Wilson quotients.

Academic career

Agoh received both his bachelor's degree and doctorate from the Tokyo University of Science, where he has been affiliated for over five decades.

His early research, emerging in the late 1970s, focused on the First Case of Fermat's Last Theorem. In a 1982 paper in the Journal of Number Theory, he investigated the relationship between Bernoulli numbers and the criteria for solutions to Fermat's equation. Over time, his work expanded to cover combinatorial identities, p-adic analysis, and recurrence relations for special sequences including Euler, Genocchi, and Stirling numbers.

Collaborations

His most prolific research partnership has been with Karl Dilcher of Dalhousie University, with whom he has co-authored numerous papers on convolution identities, reciprocity formulas, and properties of Stirling numbers. Other collaborators include Ladislav Skula, with whom he investigated Wilson quotients for composite moduli, as well as Kenichi Mori and Tetsuya Taniguchi.

Mid-2020s research

Agoh remains an active researcher as of 2025. His most recent work, published in the journal Integers, discusses natural generalisations of Fermat's congruence to composite moduli and applies these results to new characterisations of twin primes and Sophie Germain primes.

The Agoh–Giuga conjecture

The Agoh–Giuga conjecture is an unproven statement in number theory that establishes a necessary and sufficient condition for a positive integer to be prime. It results from the synthesis of two independent formulations: one proposed by the Italian mathematician Giuseppe Giuga in 1950, and one by Takashi Agoh in 1990.

Formulations

The conjecture states that a positive integer $n\ge 2$ is prime if and only if it satisfies one of the following equivalent congruences:

• Giuga formulation (1950): An integer $n$ is prime if and only if:

$\sum _{i=1}^{n-1}{i}^{n-1}\equiv -1(\mathrm{mod}n)$

• Agoh formulation (1990): An integer $n$ is prime if and only if:

$n{B}_{n-1}\equiv -1(\mathrm{mod}n)$

In the Bernoulli formulation, the residues of the rational numbers are taken in a modular sense yielding integers; this is possible because the denominators of Bernoulli numbers are square-free and therefore invertible modulo $n$.

Structural equivalence

In a 1995 paper in Manuscripta Mathematica, Agoh proved that his formulation is mathematically equivalent to Giuga's original power-sum conjecture. This proof relied on the Clausen–von Staudt theorem, which characterises the denominators of Bernoulli numbers. Agoh demonstrated that the behaviour of $n{B}_{n-1}$ modulo $n$ mirrors the behaviour of the power sum $S_{n-1}(n)$ for all odd integers, and that even composite numbers are systematically excluded from satisfying either congruence.

Anatomy of a counterexample

A composite number $n$ that satisfies the primality criteria of the conjecture is known as a strong Giuga number. For such a counterexample to exist, it must simultaneously belong to two rare classes of integers:

• Giuga numbers: Composite square-free integers where every prime divisor $p$ satisfies $p\mid (n/p-1)$.
• Carmichael numbers: Composite square-free integers satisfying $a^{n-1}\equiv 1(\mathrm{mod}n)$ for all $a$ coprime to $n$.

Any potential counterexample must be odd, square-free, and have at least nine prime factors.

Computational status

Because an analytical proof has remained elusive, researchers have used algorithms to establish lower bounds for any potential counterexample:

• In 1950, Giuga established a lower bound of $10^{1,000}$ decimal digits.
• In 1985, Bedocchi raised the bound to $10^{1,700}$ digits.
• In 1996, the Borwein team raised the bound to at least 13,800 decimal digits.
• As of 2013, using tree-based prime factor exclusion and parallel computing, the bound was raised to at least 19,908 decimal digits, with any counterexample requiring at least 4,771 distinct prime factors.

Generalisations

The conjecture has been generalised into broader contexts, including arithmetic derivatives—where a Giuga number $n$ satisfies the equation $n^{\prime }=an+1$ for some integer $a$—and into the theory of ideals in number rings. In 2025, Agoh published work exploring generalised Fermat congruences applied to composite moduli, deriving new primality conditions for twin primes and Sophie Germain primes.

Selected research contributions

Shortened recurrence relations

Agoh established shortened (or incomplete) recurrence relations for Bernoulli numbers, Euler polynomials, Genocchi numbers, and Stirling numbers. A notable feature of these identities is that many intermediate terms are absent, simplifying the calculation of higher-order values. His work in this area includes polynomial analogues of the Saalschütz–Gelfand identity and von Ettingshausen–Stern's formula.

Convolution identities and reciprocity formulas

A significant portion of Agoh's publication record concerns convolution identities of arbitrary orders, involving sums of products of Bernoulli and Euler polynomials. He generalised non-linear recurrence relations such as Miki's identity and the Matiyasevich identity to Bernoulli polynomials, extending them into bivariate and trivariate forms. He has also developed reciprocity relations in which different sums of special numbers are shown to be related symmetrically, often utilising Stirling numbers as a mathematical bridge.

Determinantal expressions

In 2019, Agoh established determinantal expressions for Bernoulli polynomials and Euler polynomials by utilising polynomial analogues of the Saalschütz–Gelfand identity. He proved that these polynomials can be expressed as determinants of lower Hessenberg-type matrices defined by alternating binomial coefficients, allowing for recursive calculation without traditional generating functions.

For an integer $n\ge 2$, Agoh derived the identity:

$B_{2n}(x)=\frac{2\det {\tilde{P}}_{n-1}(x)}{(n-1)!}$

where ${\tilde{P}}_{n-1}(x)$ is a lower triangular matrix whose entries are determined by recurrence relations excluding many intermediate terms. Agoh also applied this framework to locate non-trivial zeros of Bernoulli polynomials, confirming that the golden ratio $\phi$ satisfies $B_{11}(\phi )=0$.

Fermat and Wilson quotients for composite moduli

In collaboration with Karl Dilcher and Ladislav Skula, Agoh extended the study of Fermat and Wilson quotients—classically defined for prime moduli—to composite moduli $m\ge 2$. This research established Wilson-type theorems for composite moduli and has relevance to cryptographic applications where such quotients are used to construct pseudo-random sequences.

Twin primes and Sophie Germain primes

In 2025, Agoh published research on natural generalisations of Fermat's congruence to composite moduli. He established that for integers $n,m$ with $\gcd (n,m)=1$, the combined congruence:

$a^{m-1}\equiv \frac{a^{m-n}-1}{m-n}\cdot n+a^{m-n}(\mathrm{mod}nm)$

holds if and only if both $n$ and $m$ are either prime or Carmichael numbers. He applied these generalised congruences to derive new characterisations for twin primes and Sophie Germain primes, establishing that a prime $p$ is a Sophie Germain prime if and only if, for every integer $a$ coprime to $p(2p+1)$:

$a^{2p}\equiv (2p+1)a^{p+1}-2p(\mathrm{mod}p(2p+1))$

Bernoulli polynomials and the golden ratio

In his 2019 paper in Integers, Agoh provided a formal proof that the golden ratio $\phi =\frac{1+\sqrt{5}}{2}$ and its conjugate are roots of $B_{11}(x)$:

$B_{11}(\phi )=B_{11}(\overline{\phi })=0$

He further demonstrated that all values of even-index Bernoulli polynomials at $x=\phi$ are rational (i.e., $B_{2n}(\phi )\in \mathbb{\mathbb{Q}}$ for all $n\ge 0$), and generalised this result: if ${\theta }_b$ is a root of the quadratic equation $x^2-x=b$ for any $b\in \mathbb{\mathbb{Q}}$, then $B_{2n}({\theta }_b)$ is always rational.

Giuga and Carmichael sequences

Giuga sequences

A finite increasing sequence of integers $[n_1,\dots ,n_m]$ is defined as a Giuga sequence if it satisfies the sum-minus-product condition:

$\sum _{i=1}^m\frac{1}{n_i}-\prod _{i=1}^m\frac{1}{n_i}\in \mathbb{\mathbb{N}}$

This is equivalent to requiring that $n_i\mid (n/n_i-1)$ for every element, where $n$ is the product of all elements. There are infinitely many Giuga sequences of any length, though they typically contain even factors.

Carmichael sequences

Carmichael sequences are defined as finite increasing sequences $[a_1,\dots ,a_m]$ where $(a_i-1)\mid (n/a_i-1)$ for all $i$. Giuga's conjecture would be proven if it could be shown that no Giuga sequence can simultaneously be a Carmichael sequence.

Relation to Wilson's theorem

Theoretical treatments of the Agoh–Giuga conjecture note its complementary relationship with Wilson's theorem. While Wilson's theorem provides a multiplicative characterisation of primes—$(p-1)!\equiv -1(\mathrm{mod}p)$—the Agoh–Giuga conjecture provides an additive one. Combined, the two imply that a number $p$ is prime if and only if it satisfies both the power-sum condition and the factorial product condition simultaneously.

Relationship with arithmetic derivatives

The theory of Giuga numbers and the Agoh–Giuga conjecture has been enriched by the study of the arithmetic derivative, denoted $n^{\prime }$. The arithmetic derivative is defined by the Leibniz rule $(ab{)}^{\prime }=a^{\prime }b+a{b}^{\prime }$, with $p^{\prime }=1$ for any prime $p$.

For a square-free integer $n=p_1{p}_2\cdots p_r$, the derivative is:

$n^{\prime }=n\sum _{i=1}^r\frac{1}{p_i}$

A central result in this area establishes that a positive integer $n$ is a Giuga number if and only if:

$n^{\prime }=an+1$

for some natural number $a\in \mathbb{\mathbb{N}}$. It has been conjectured (Lava's conjecture) that $a$ must always equal 1 for Giuga numbers.

Cyclotomic fields and Kummer-type congruences

Agoh's early research focused on the arithmetic of cyclotomic fields and the First Case of Fermat's Last Theorem. In the late 1970s and 1980s, he published on the relationship between Bernoulli numbers and Kummer-type congruences, investigating criteria for the existence of solutions to Fermat's equation through Voronoi and Kummer congruences. He also utilised systems of congruences to explore the bases of Stickelberger subideals in certain group rings, a technique used in the study of $p$-adic functions and cyclotomic units.

Strong Giuga numbers and prime properties

Agoh's 1995 reformulation proved that a composite integer $n$ satisfying the primality congruence must simultaneously be a Giuga number and a Carmichael number. This rare class is referred to in technical literature as strong Giuga numbers. Because all Carmichael numbers are odd, any potential counterexample to the conjecture must also be an odd integer.

For a composite number to satisfy these restrictive criteria, every prime divisor $p$ must fulfil the condition $p^2(p-1)\mid (n-p)$. Current research indicates that if an odd Giuga number exists, it must be the product of at least 14 distinct prime factors.

Selected bibliography

• Agoh, Takashi (1982). "On Fermat's last theorem and the Bernoulli numbers". Journal of Number Theory 14 (2): 149–156. doi:10.1016/0022-314X(82)90061-0. 
• Agoh, Takashi (1995). "On Giuga's conjecture". Manuscripta Mathematica 87 (4): 501–510. doi:10.1007/bf02570490. 
• Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli". Mathematics of Computation 67 (222): 843–861. doi:10.1090/S0025-5718-98-00952-0. 
• Agoh, Takashi; Dilcher, Karl (2008). "Reciprocity Relations for Bernoulli Numbers". American Mathematical Monthly 115 (3): 237–244. 
• Agoh, Takashi (2014). "Convolution Identities for Bernoulli and Genocchi Polynomials". Electronic Journal of Combinatorics 21 (1). https://www.combinatorics.org/. 
• Agoh, Takashi; Dilcher, Karl (2014). "Higher-order convolutions for Bernoulli and Euler polynomials". Journal of Mathematical Analysis and Applications 419 (2): 1235–1247. doi:10.1016/j.jmaa.2014.05.043. 
• Agoh, Takashi (2017). "Shortened recurrence relations for generalized Bernoulli polynomials". Journal of Number Theory 176: 149–173. doi:10.1016/j.jnt.2016.12.012. 
• Agoh, Takashi (2018). "On Shortened Recurrence Relations for Genocchi Numbers and Polynomials". Integers 18 (A70). https://math.colgate.edu/~integers/s70/s70.pdf. 
• Agoh, Takashi (2019-02-01). "Determinantal Expressions for Bernoulli Polynomials". Integers 19 (A9). https://math.colgate.edu/~integers/t9/t9.pdf. 
• Agoh, Takashi (2020). "On Bivariate and Trivariate Miki-type Identities for Bernoulli Polynomials". Integers 20 (A23). 
• Agoh, Takashi (2025-03-26). "A Note on Fermat's Congruence". Integers 25 (A31). doi:10.5281/zenodo.15091101. https://math.colgate.edu/~integers/z31/z31.pdf. 

See also

• Bernoulli number
• Carmichael number
• Giuga number
• Korselt's criterion
• Wilson's theorem

External links

• Agoh's Conjecture at MathWorld
• Takashi Agoh author profile at zbMATH Open
• Takashi Agoh at dblp computer science bibliography