Giuseppe Melfi

Key takeaways

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Reference excerpt

Giuseppe Melfi (June 11, 1967) is an Italo-Swiss mathematician who works on practical numbers and modular forms.

Career He gained his PhD in mathematics in 1997 at the University of Pisa. After some time spent at the University of Lausanne during 1997-2000, Melfi was appointed at the University of Neuchâtel, 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 practical numbers. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of primes. 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 {\displaystyle m-2,m,m+2} . He also proved that there exist infinitely many Fibonacci numbers that are practicals Another notable contribution has been in an application of the theory of modular forms, 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

∑ k ≡ 1 mod 3 0 < k < n σ ( k ) σ ( n − k ) = 1 9 σ 3 ( n ) for n ≡ 2 mod 3 {\displaystyle \sum _{\stackrel {0<k<n}{k\equiv 1{\bmod {3}}}}\sigma (k)\sigma (n-k)={\frac {1}{9}}\sigma _{3}(n)\qquad {\mbox{ for }}n\equiv 2{\bmod {3}}}

where σ ( n ) {\displaystyle \sigma (n)} is the sum of the divisors of n {\displaystyle n} and σ 3 ( n ) {\displaystyle \sigma _{3}(n)} is the sum of the third powers of the divisors of n {\displaystyle n} . In 1996 Paul Erdős wrote him two letters in which several problems were proposed. They met in Eger in July

, but Erdős died in September. In the subsequent decade Melfi studied some Erdős problems with some notable contribution in particular concerning sum-free sequences. Among other problems in elementary number theory, 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. Since 2019, with a 14712-digits primitive weird number, he shares this record, with a team of collaborators. In applied mathematics his research interests include probability and simulation.

Selected research publications Giuseppe Melfi and Yadolah Dodge (2008). Premiers pas en simulation. Springer-Verlag. ISBN 978-2-287-79493-3. Deshouillers, Jean-Marc; Erdős, Pál; Melfi, Giuseppe (1999), "On a question about sum-free sequences", Discrete Mathematics, 200 (1–3): 49–54, doi:10.1016/s0012-365x(98)00322-7, MR 1692278. Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory. 147. Elsevier: 508–514. doi:10.1016/j.jnt.2014.07.024. Melfi, Giuseppe (1996), "On two conjectures about practical numbers", Journal of Number Theory, 56 (1): 205–210, doi:10.1006/jnth.1996.0012, MR 1370203

Illustrations

Worked examples

Example 1 — a first encounter with Giuseppe Melfi

Start with the simplest possible case. Write down what Giuseppe Melfi claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Giuseppe Melfi before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Giuseppe Melfi ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Giuseppe Melfi

In research: Giuseppe Melfi appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Giuseppe Melfi in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Giuseppe Melfi is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1967 births, 20th-century Italian mathematicians, 20th-century Swiss mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Giuseppe Melfi outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Giuseppe Melfi in 20 minutes

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Frequently asked questions

What is Giuseppe Melfi in simple terms?

Giuseppe Melfi (June 11, 1967) is an Italo-Swiss mathematician who works on practical numbers and modular forms. Career He gained his PhD in mathematics in 1997 at the University of Pisa.

Why does Giuseppe Melfi matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Giuseppe Melfi?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Giuseppe Melfi.