Giuseppe Melfi (June 11, 1967) is an Italo-

Swiss Swiss most commonly refers to: * the adjectival form of Switzerland * Swiss people Swiss may also refer to: Places * Swiss, Missouri * Swiss, North Carolina * Swiss, West Virginia * Swiss, Wisconsin Other uses * Swiss Café, an old café located ...

mathematician who works on

practical number In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 ...

s and

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

Career

He gained his PhD in mathematics in 1997 at the

University of Pisa The University of Pisa (, UniPi) is a public university, public research university in Pisa, Italy. Founded in 1343, it is one of the oldest universities in Europe. Together with Scuola Normale Superiore di Pisa and Sant'Anna School of Advanced S ...

. After some time spent at the

University of Lausanne The University of Lausanne (UNIL; ) in Lausanne, Switzerland, was founded in 1537 as a school of Protestant theology, before being made a university in 1890. The university is the second-oldest in Switzerland, and one of the oldest universities ...

during 1997-2000, Melfi was appointed at the

University of Neuchâtel The University of Neuchâtel (UniNE) is a French-speaking public research university in Neuchâtel, Switzerland. The university has four faculties (schools) and more than a dozen institutes, including arts and human sciences, natural sciences, ...

, as well as at the University of Applied Sciences Western Switzerland and at the local University of Teacher Education.

Work

His major contributions are in the field of

s. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of

primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. Melfi proved two conjectures both raised in 1984: the first one states that every even number is a sum of two practical numbers, a property that is still unproven for primes. The second one is that there exist infinitely many triples of practical numbers of the form

m-2,m,m+2

. He also proved that there exist infinitely many

Fibonacci numbers In mathematics, the Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many writers begin the s ...

that are practicals Another notable contribution has been in an application of the theory of

modular forms In mathematics, a modular form is a holomorphic function on the Upper half-plane#Complex plane, complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the Group action (mathematics), group action of the ...

, where he found new Ramanujan-type identities for the sum-of-divisor functions. His seven new identities extended the ten other identities found by Ramanujan in 1913. In particular he found the remarkable identity :

\sum_ \sigma(k)\sigma(n-k)=\frac19\sigma_3(n) \qquad \mboxn\equiv2\bmod3

where

\sigma(n)

is the sum of the divisors of

n

and

\sigma_3(n)

is the sum of the third powers of the divisors of

n

. Among other problems in elementary

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, he is the author of a theorem that allowed him to get a 5328-digit number that has been for a while the largest known primitive

weird number In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divi ...

. Since 2019, with a 14712-digits primitive weird number, he shares this record, with a team of collaborators.Amato, G. Hasler, M.F., Melfi, G. and Parton, M. ''Primitive abundant and weird numbers with many prime factors'', Journal of number theory 201 (2019), 436-459. In applied mathematics his research interests include

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

and

simulation A simulation is an imitative representation of a process or system that could exist in the real world. In this broad sense, simulation can often be used interchangeably with model. Sometimes a clear distinction between the two terms is made, in ...

Selected research publications

* *. * *

References

External links

Giuseppe Melfi's home page

The proof of conjectures on practical numbers
an
the joint work with Paul Erdős
on Zentralblatt.
Tables of practical numbers
compiled by Giuseppe Melfi
Academic research query for "Giuseppe Melfi"
{{DEFAULTSORT:Melfi, Giuseppe 1967 births 20th-century Italian mathematicians 21st-century Italian mathematicians Living people Number theorists Mathematicians from Sicily Academic staff of the University of Neuchâtel

Career

Work

Selected research publications

See also

References

External links