Top Five Dutch Mathematicians

Identifying the top five Dutch mathematicians involves evaluating their contributions to mathematics, focusing on impact, innovation, and recognition within the field. The Netherlands has a rich mathematical tradition, producing scholars whose work has shaped various areas of mathematics. Based on historical significance, influence, and contributions, the following are five Dutch mathematicians who could be considered among the top, presented in no strict order of ranking due to the subjective nature of "top" and the diverse areas of their work.

Top Five Dutch Mathematicians

1. Christiaan Huygens (1629–1695)

Contributions: Huygens was a polymath whose mathematical work significantly advanced geometry, probability, and applied mathematics. He developed the wave theory of light, which involved mathematical modeling of wave propagation, and made foundational contributions to probability theory through his work on games of chance, including the concept of expected value in De Ratiociniis in Ludo Aleae (1657). He also improved methods for calculating the areas of curves and surfaces, contributing to early calculus. His work on the pendulum clock involved mathematical analysis of periodic motion, impacting physics and mathematics.  

Significance: Huygens is considered a pioneer in probability and a key figure in the Scientific Revolution. His mathematical insights bridged theory and application, influencing later developments in calculus and physics.  

Why Included: His foundational work in probability and geometry, combined with his international influence, makes him a standout Dutch mathematician.

https://en.wikipedia.org/wiki/Christiaan_Huygens

(Image: http://ressources2.techno.free.fr/informatique/sites/inventions/inventions.html)

2. Luitzen Egbertus Jan Brouwer (1881–1966)

Contributions**: Brouwer founded intuitionism, a philosophy of mathematics emphasizing constructive proofs over classical logic. He developed key results in topology, including the Brouwer Fixed-Point Theorem, which states that any continuous function from an \( n \)-dimensional closed ball to itself has a fixed point. His work on the foundations of mathematics challenged the use of the law of the excluded middle and influenced modern topology and mathematical logic.  

 Significance: The Brouwer Fixed-Point Theorem is a cornerstone of topology, with applications in economics, game theory, and physics. Intuitionism reshaped debates in mathematical philosophy, making Brouwer a transformative figure.  

Why Included: His groundbreaking contributions to topology and philosophy of mathematics cement his place as one of the most influential Dutch mathematicians.

https://en.wikipedia.org/wiki/L._E._J._Brouwer

(Image:  http://www-history.mcs.st-andrews.ac.uk/Biographies/Brouwer.html)

3. Willem Klein (1911–1986)  

Contributions: Known as “Wim Klein, the Human Computer,” he was a mental calculator who performed complex computations, such as finding roots and factorizations, with extraordinary speed. While not a theoretical mathematician, his work at CERN and the University of Amsterdam involved practical applications of mathematical computation, including programming and numerical analysis, during the early computer era.  

 Significance: Klein’s computational feats popularized mathematics and bridged human calculation with emerging computer science. His work at CERN contributed to scientific computations in physics.  

Why Included: His unique contribution as a computational prodigy and his role in applied mathematics at a pivotal time in technology make him a notable figure, though less theoretical than others.

https://en.wikipedia.org/wiki/Willem_Klein

Wim Klein shows off his skills during a public lecture at CERN (Image: CERN)

4. Hans Freudenthal (1905–1990)

 Contributions: Freudenthal made significant contributions to algebraic topology, Lie groups, and mathematics education. He developed the Freudenthal Suspension Theorem, which describes the stabilization of homotopy groups, a key result in topology. In education, he founded the “realistic mathematics education” approach, emphasizing intuitive and practical learning, influencing global mathematics curricula.  

Significance: His topological work advanced the understanding of homotopy theory, while his educational reforms transformed how mathematics is taught, particularly in the Netherlands.  

Why Included: His dual impact in pure mathematics and education highlights his versatility and lasting influence.

https://en.wikipedia.org/wiki/Hans_Freudenthal

(Attribution: L.H. Hofland / collectie Het Utrechts Archief)

5. Jan Hogendijk (born 1955)

Contributions: A contemporary mathematician and historian of mathematics, Hogendijk specializes in the history of Islamic mathematics and its transmission to Europe. His work includes detailed studies of medieval mathematicians like Al-Khwārizmī and the geometry of the Banū Mūsā brothers, as well as contributions to the history of Greek and Dutch mathematics.  

Significance: While not a “pure” mathematician in the sense of developing new theorems, his rigorous analyses of historical mathematical texts have clarified the development of algebra and geometry, earning international recognition.  

Why Included: His scholarly work bridges mathematics and its historical context, making him a prominent modern Dutch mathematician.

https://en.wikipedia.org/wiki/Jan_Hogendijk

(Image: https://www.jphogendijk.nl/)

These five Dutch mathematicians—Huygens, Brouwer, Klein, Freudenthal, and Hogendijk—represent a range of contributions from probability and topology to education and historical scholarship, showcasing the Netherlands’ mathematical heritage.

(Note: This post was Grok-assisted.)

Narcissistic Numbers

 (Image: Narciso (Jan Cossiers), CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons)

In Greek mythology, Narcissus was known for his beauty. He rejected advances from all suitors, both men and women, and instead fell in love with his own reflection. The legend gives us the source of the words narcissist and narcissistic. 

There are also numbers known as Narcissistic Numbers. In the On-Line Encyclopedia of Integer Sequences (OEIS), sequence A005188 defines such numbers to be:  m-digit positive numbers equal to the sum of the m-th powers of their digits.

For example, the number 153 has 3 digits. The sum of 1^3 + 5^3 + 3^3 = 153. The sequence of Narcissistic Numbers begins: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407...

Narcissistic Numbers.

Münchhausen Numbers

 

(Image: Statue of Baron Münchhausen riding half of a horse - https://commons.wikimedia.org/wiki/User:Franzfoto)

Baron Münchhausen is a fictional character known for making exaggerated stories (other spellings include Munchhausen and Munchausen). In addition to the fables of the Baron, the name is used to describe Münchhausen syndrome and Münchhausen syndrome by proxy, in which someone fabricates a story of their illness or the illness of one they care for to gain attention or sympathy.

There are also numbers known as Münchhausen Numbers. These are numbers that equal the sum of each of their digits raised to their own power. For example, in base ten, the numbers include 1 and 3435. 1^1=1, 3^3 +4^4+3^3+5^5=3435. This sequence of numbers is A046253 in the On-Line Encyclopedia of Integer Sequences (OEIS). Another sequence of other Münchhausen Numbers is A166623, which lists such numbers in multiple bases.

Achilles Numbers

 

(Photo: Closeup of Achilles thniskon in Corfu Achilleion  - Dr.K., CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons)

In Greek mythology, Achilles, a central figure in Homer's Iliad, was known as the greatest Greek warrior. By legend, Achilles was invulnerable in all of his body except for one heel. According to myth, when his mother Thetis dipped him in the river Styx as an infant, she held him by one of his heels, leaving it untouched by the waters and thus his only vulnerable body part. To this day, Achilles' heel refers to a point of weakness that can lead to downfall.

There is also a series of numbers named for Achilles. Sequence A052486 of the On-Line Encyclopedia of Integer Sequences (OEIS) describes Achilles numbers as "powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power."

This definition uses the names of other types of numbers. 

Powerful numbers:  if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers). This is sequence A001694 in the OEIS.

Perfect numbers: Numbers k such that k is equal to the sum of the proper divisors of k. See OEIS: A000396.

Perfect power: Numbers of the form m^k where m > 0 and k >= 2.  See OEIS: A001597.

The first few Achilles numbers are: 72, 108, 200, 288, 392.

This post was inspired by the comment left by a reader to a recent post about the number 108. He recognized that 108 is also an Achilles number.

108: A very interesting number

(Image: Grok)

I asked Grok to write a post about the number 108:

The Magic of the Number 108

The number 108 may seem unremarkable at first glance, but dig a little deeper, and you’ll uncover a treasure trove of mathematical beauty and cultural significance. From its prime factorization to its role in number theory and beyond, 108 is a number that bridges the abstract world of mathematics with the tangible world of human experience. Let’s dive into what makes 108 so special!

Prime Factorization and Divisors

To understand 108 mathematically, let’s start with its prime factorization:

[ 108 = 2^2 *3^3 ]

This tells us that 108 is built from two prime numbers: 2 (squared) and 3 (cubed). The exponents give us insight into its structure, and we can use them to find the number of divisors using the divisor function. For a number ( n = p_1^{e_1} * p_2^{e_2}  ), the number of divisors is:

[ (e_1 + 1)(e_2 + 1)  ]

For 108, with ( 2^2 ) (exponent 2) and ( 3^3 ) (exponent 3), we calculate:

[ (2 + 1)(3 + 1) = 3*4 = 12 ]

So, 108 has 12 divisors. Let’s list them:

[ 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 ]

This abundance of divisors makes 108 a highly composite number—a number with more divisors than any smaller positive integer. While 108 isn’t the smallest highly composite number (that’s 12, with 6 divisors), its divisor count contributes to its versatility in mathematical contexts.

Sum of Divisors and Number Theory

The sum of the divisors of 108 (including 108 itself) can be computed using the sum-of-divisors function, ( sigma(n) ). For ( n = p_1^{e_1}*p_2^{e_2} ), we use:

[ sigma(n) = sigma(p_1^{e_1})*sigma(p_2^{e_2}) ]

where ( sigma(p^e) = 1 + p + p^2 +...+ p^e ). For 108:

• For ( 2^2 ): ( sigma(2^2) = 1 + 2 + 4 = 7 )
• For ( 3^3 ): ( sigma(3^3) = 1 + 3 + 9 + 27 = 40 )

Thus:

[ sigma(108) = 7 *40 = 280 ]

Excluding 108 itself, the sum of the proper divisors is:

[ 280 - 108 = 172 ]

Since 172 is greater than 108, 108 is an abundant number, meaning its proper divisors sum to more than the number itself. Interestingly, 108 is not a perfect number (where the sum equals the number, like 28) or a deficient number (where the sum is less, like 16).

2-Valuation of 108

The 2-valuation of a number is the highest power of 2 that divides it. For 108, we look at the prime factorization ( 2^2 * 3^3 ). The exponent of 2 is 2, so the 2-valuation of 108 is:

[ \nu_2(108) = 2 ]

This means ( 2^2 = 4 ) is the largest power of 2 that divides 108 evenly. In Mathematica, you can compute this with:

IntegerExponent[108, 2]
(* Output: 2 *)

The 2-valuation is a key concept in number theory, especially in areas like p-adic analysis.

Anti-Divisors of 108

Anti-divisors are numbers ( k < n ) such that ( n ) divides ( 2k ), but ( k ) does not divide ( n ). To find the anti-divisors of 108, we can use the following Mathematica expression:

Select[Range[1, 107], Divisible[2*#, 108] && !Divisible[108, #] &]
(* Output: {9, 18, 27, 36, 54} *)

Let’s verify one: For ( k = 9 ), ( 2k = 18 ), and ( 108 \div 18 = 6 ), an integer, so 108 divides 18. Also, ( 108 \div 9 = 12 ), not an integer, so 9 does not divide 108. Thus, 9 is an anti-divisor. The anti-divisors of 108 are:

[ {9, 18, 27, 36, 54} ]

These numbers are intriguing because they reveal a complementary relationship to the divisors, highlighting 108’s structure in a less obvious way.

Cultural and Mathematical Significance

Beyond its number-theoretic properties, 108 holds a special place in various cultural and mathematical contexts:

• Geometry: 108 degrees is the interior angle of a regular pentagon. This connects 108 to the golden ratio and pentagonal symmetry.
• Astronomy: The number 108 appears in approximate ratios involving celestial bodies. For example, the distance from Earth to the Sun is about 108 times the Sun’s diameter, and the distance to the Moon is about 108 times the Moon’s diameter (though these are rough approximations).
• Cultural Contexts: In Hinduism and Buddhism, 108 is considered sacred. There are 108 beads in a mala (prayer necklace), 108 Upanishads, and 108 sacred sites in some traditions. While these are not strictly mathematical, they suggest 108’s universal appeal, perhaps due to its mathematical richness.

Fun Fact: 108 in Sequences

The number 108 appears in various mathematical sequences. For example, it’s a tetrahedral number, representing the sum of the first 8 triangular numbers:

[ 108  = 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 ]

It’s also the hyperfactorial of 3, defined as ( 1^1 *2^2 * 3^3 = 1 * 4 * 27 = 108 ). These appearances make 108 a recurring star in combinatorial mathematics.

Conclusion

The number 108 is a fascinating blend of mathematical elegance and cultural resonance. Its prime factorization, abundant divisors, 2-valuation, and anti-divisors reveal a rich structure that invites exploration. Whether you’re a number theory nerd or just curious about the world of numbers, 108 offers something to marvel at. So, the next time you encounter 108—whether in a math problem, a prayer bead, or a pentagon’s angle—take a moment to appreciate its hidden magic!

Have you encountered 108 in an interesting mathematical context? Share your thoughts in the comments below, and let’s keep the number love going!

(Above post written by Grok and inspired by the recent article in The Guardian by Alex Bellos.

50400: A Highly Composite Number

This blog recently had its 50400th visitor. The number 50400 is the 27th highly composite number. A highly composite number has more divisors than all previous highly composite numbers. In the case of 50400, it has 108 divisors.

The first 27 highly composite numbers and the number of their divisors are:

Order	Highly Composite Number	Number of Divisors
1	1	1
2	2	2
3	4	3
4	6	4
5	12	6
6	24	8
7	36	9
8	48	10
9	60	12
10	120	16
11	180	18
12	240	20
13	360	24
14	720	30
15	840	32
16	1260	36
17	1680	40
18	2520	48
19	5040	60
20	7560	64
21	10080	72
22	15120	80
23	20160	84
24	25200	90
25	27720	96
26	45360	100
27	50400	108

The highly composite numbers are designated in the On-Line Encyclopedia of Integer Sequences as A002182 , and the number of their divisors is the sequence A002183.