Mersenne primes are a special class of prime numbers that can be expressed in the form \( M_n = 2^n - 1 \), where \( n \) is a positive integer. For a number of this form to be classified as a Mersenne prime, \( n \) itself must also be a prime number. The reason for this restriction is that if \( n \) is composite (i.e.

A prime gap is the difference between two successive prime numbers. For example, if \( p_n \) is the \( n \)-th prime number, then the prime gap \( g_n \) between the \( n \)-th and the \( (n+1) \)-th prime can be expressed as: \[ g_n = p_{n+1} - p_n \] Prime gaps can vary significantly in size.

The reciprocal of a prime number is defined as \( \frac{1}{p} \), where \( p \) is a prime number. Primes are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, 13, and so on.

"A Treatise on Probability" is a foundational work on probability theory written by the British mathematician and philosopher, John Maynard Keynes, first published in 1921. In this treatise, Keynes explores the mathematical framework of probability and its philosophical implications. The work is known for its systematic approach to the theory of probability, discussing its application in various fields, including economics.

The Electronic Journal of Probability (EJP) is an academic journal that focuses on research in the field of probability theory. It publishes articles that cover a wide range of topics within probability, including but not limited to theoretical developments, applications, and connections to other areas of mathematics. EJP is known for its open-access model, meaning that all articles published in the journal are freely accessible to anyone, promoting the dissemination of knowledge and facilitating collaboration among researchers.

Anatoliy Skorokhod is a prominent figure in the field of mathematics, particularly known for his contributions to the theory of stochastic processes and mathematical statistics. He is recognized for developing key concepts and results in areas such as stochastic integration, the theory of random processes, and functional limit theorems. His work often bridges the gap between theoretical mathematics and its applications in various fields, including finance, physics, and engineering.

"Stochastic Processes and Their Applications" generally refers to both the theory and practical applications of stochastic processes, which are mathematical objects used to describe systems that evolve over time in a probabilistic manner. A stochastic process is a collection of random variables representing a process that evolves over time, where the next state of the process is influenced by its current state and possibly also by some random noise or other external factors.

Allan Sly is an American mathematician known for his work in probability theory, particularly in statistical physics and combinatorial structures. He has contributed significantly to the understanding of phase transitions and the behavior of random processes. Sly is notably recognized for his work on the "hard core model" and the "random cluster model," after proving the conjecture regarding the transition between uniqueness and non-uniqueness of the measures for these models.

Boris Tsirelson is a notable Russian mathematician, primarily recognized for his contributions to the fields of functional analysis, probability theory, and quantum mechanics. He is particularly known for his work related to Bell's theorem and non-locality in quantum mechanics. One of his significant contributions is the construction of what are now known as Tsirelson spaces, which are certain types of Banach spaces that arise in the study of quantum probabilities.

Boris Vladimirovich Gnedenko (1912–1995) was a prominent Soviet mathematician known for his contributions to probability theory and mathematical statistics. He made significant contributions to various areas within these fields, including the study of queues, reliability theory, and stochastic processes. One of his notable achievements was his work on the theory of extreme values, which has implications in various practical applications such as engineering and quality control.

Chris Rogers is a well-known mathematician primarily recognized for his work in probability theory and its connections to various fields, including statistical mechanics and stochastic processes. He has contributed significantly to areas such as random walks, large deviations, and Brownian motion. Rogers has authored and co-authored several influential papers and books in mathematics. In addition to his research contributions, he has also been involved in academic teaching and mentoring, influencing many students and researchers in the field.

Edward W. Piotrowski is not a widely recognized public figure, so there may not be substantial information readily available about him in popular databases or general knowledge sources. He could be a private individual, a professional in a specific field, or perhaps someone involved in local matters or organizations not covered extensively in major media.

Erol Gelenbe is a prominent Turkish computer scientist and engineer known for his contributions to the fields of computer networks, computer systems, and performance evaluation. He has conducted significant research in areas such as queuing theory, network design, and the application of artificial intelligence. Gelenbe is also recognized for his work on various theoretical models and algorithms that have practical applications in networking and telecommunications. He has held academic positions at various institutions and has published numerous research papers in his fields of expertise.

Friedrich Götze may refer to multiple individuals, but one notable figure with that name was a German artist and painter known for his work in the late 19th and early 20th centuries. However, there isn't a substantial amount of information widely available about him. It's worth noting that there could be other individuals by that name in different fields or contexts.

John Muth is known for his contributions to economics, particularly in the fields of finance and behavioral economics. He is best known for his development of the Rational Expectations Hypothesis, which suggests that individuals form their expectations about the future based on all available information, and that these expectations are, on average, correct. Muth's work has had a significant impact on economic theory and has influenced various areas such as macroeconomics and financial markets.

Greg Lawler is a prominent mathematician known for his work in probability theory and mathematical physics. He is particularly recognized for his contributions to the field of random walks, percolation theory, and the study of two-dimensional critical phenomena. Lawler has also written influential texts on the subject, including his work on the intersection of probability theory and complex analysis. In addition to his research, he has been involved in education and has held academic positions at various institutions.

Grégory Miermont does not appear to be a widely recognized public figure or a notable individual based on available information as of October 2023. It is possible that he is a private individual, a professional in a specific field, or a name that has emerged after my last update.

Hans Reichenbach (1891–1953) was a prominent German philosopher and a key figure in the development of logical positivism and the philosophy of science. He was associated with the Berlin Circle, a group of philosophers and scientists that sought to synthesize scientific knowledge with logical analysis. Reichenbach's work focused on the philosophy of space and time, the theory of probability, and sciences like physics and epistemology.

Jacob Wolfowitz is a noted figure in the fields of statistics and economics. He is particularly known for his contributions to the field of statistical decision theory and for his work on the foundations of statistical inference. One of the notable concepts associated with his name is the **Wolfowitz criterion** for statistical decision-making, which relates to hypothesis testing and provides criteria for making decisions based on statistical data. He also contributed to areas such as Bayesian statistics and graphical models.

János Galambos is a notable figure primarily recognized in the field of mathematics, particularly in combinatorics and graph theory. He has made significant contributions to these areas through his research and publications.