Heegner's lemma is a result in number theory that is primarily concerned with the representation of integers as sums of squares. It plays an important role in the theory of quadratic forms and has implications in the study of class numbers and other aspects of algebraic number theory. Specifically, Heegner's lemma provides a condition under which certain integers can be represented as sums of two squares.

Alan Baker is a prominent mathematician known for his contributions to number theory, particularly in the areas of transcendental numbers and Diophantine equations. Born on 19 January 1939, he was awarded the Fields Medal in 1970 for his groundbreaking work in transcendental number theory, specifically for his development of methods to prove the transcendence of certain numbers.

As of my last update in October 2023, there is no widely recognized person, place, or concept specifically known as "Albert Ingham." It's possible that you are referring to a lesser-known individual, a fictional character, or a recent development that has not been included in my training data.

Alfred van der Poorten was a prominent figure in the field of mathematics, particularly known for his work in topology, geometry, and mathematical logic. He made significant contributions to various areas of mathematics and was recognized for his research and publications. Additionally, he was involved in mathematical education and served in academic positions throughout his career.

Andrew Booker is a mathematician known for his work in number theory and combinatorial mathematics. He has made contributions in areas such as prime numbers, partitions, and combinatorial algorithms. One of his notable achievements includes the development of methods related to sums of squares and the search for certain types of numbers with specific mathematical properties. In addition to his research contributions, Booker is also involved in the mathematical community through collaboration, outreach, and education.

Haruzo Hida is a prominent Japanese mathematician known for his work in number theory and algebraic geometry. He has made significant contributions to the field, particularly in relation to p-adic Hodge theory and the study of motives. Hida has worked extensively on the Hida theory of ordinary forms and modular forms, which are essential in understanding the connections between different areas of mathematics, such as representation theory and arithmetic geometry.

Daniel Shanks (1917-1996) was an American mathematician known for his work in number theory, particularly in the areas of primality testing and the computation of π (pi). He is perhaps best known for creating the Shanks-Tonelli algorithm for finding square roots modulo a prime and for his contributions to the theory of continued fractions. In addition to his research, Shanks was a prolific author, and he wrote several important mathematical papers and books.

Dimitris Koukoulopoulos is a name that does not appear to be widely recognized or associated with any particular public figure or notable event in popular culture, academia, or other well-known fields as of my last knowledge update in October 2023.

George Leo Watson (1873–1935) was a British botanist and plant pathologist known for his contributions to the study of fungi, particularly in relation to plant diseases. He is recognized for his research in mycology and for developing methods to identify and classify various fungal species that affect plants. His work has had a lasting impact on the field of plant pathology and the management of crop diseases.

Hans-Egon Richert does not appear to be a widely recognized figure in public domains such as literature, politics, science, or entertainment as of my last knowledge update in October 2023. It's possible that he is a less widely known individual or that the name could refer to someone in a niche field or context.

Hendrik Kloosterman is not widely known as a public figure or entity in popular culture or mainstream knowledge as of my last update in October 2023. It's possible that Hendrik Kloosterman could be a professional in a specific field, an academic, or a private individual who does not have significant public visibility. Could you provide more context or specify the area in which you are interested?

"Grapeshot" can refer to different concepts depending on the context: 1. **Historical Weaponry**: Grapeshot is a type of ammunition used in cannons and artillery. It consists of small, round projectiles (like balls or fragments of metal) that are packed together in a canvas bag or metal canister. When fired, it spreads out to cover a larger area, making it effective against infantry and small targets, similar to how a shotgun works.

Henry John Stephen Smith was a prominent British mathematician known for his contributions to various fields, including number theory and algebra. He was born on April 13, 1826, and died on May 30, 1883. Smith is particularly noted for his work on the theory of quadratic forms and his development of what is now known as Smith's normal form for matrices. His contributions laid foundational work that has influenced subsequent developments in these areas of mathematics.

Henryk Iwaniec is a prominent Polish mathematician known for his work in analytic number theory, particularly in areas such as the distribution of prime numbers, modular forms, and automorphic forms. He has made significant contributions to various problems in number theory, including the development of sieve methods and the study of L-functions. Iwaniec has also authored and co-authored several influential research papers and books on these subjects.

Hervé Jacquet may refer to several things, but one notable association is with the French politician Hervé Jacquet, who has been involved in local politics in France. However, without more context, it's difficult to provide a specific answer as there may also be other individuals or entities with that name.

Imre Z. Ruzsa is a Hungarian mathematician known for his contributions to number theory and combinatorial number theory. He is notably recognized for his work on additive number theory and is one of the co-authors of the well-known result called the "Ruzsa-Szemerédi Theorem," which deals with the structure of sets of integers and their additive properties.

Ivan Ivanov is a mathematician known for his contributions to various fields within mathematics, particularly in areas related to optimization, game theory, and mathematical economics. However, specific details about his academic achievements, publications, or particular contributions may vary and would be dependent on the context or area of focus, as there are several individuals named Ivan Ivanov in mathematics.

Jeffrey Hoffstein is a mathematician known for his work in number theory and algebraic geometry, as well as for his contributions to cryptography. He has been involved in research related to the mathematical foundations of cryptographic systems and has contributed to various areas including lattice-based cryptography. Hoffstein is a professor at Brown University and has authored or co-authored numerous papers and publications in his field.

John Pollard is a notable mathematician, particularly recognized for his contributions to number theory and computational mathematics. He is best known for the Pollard rho algorithm, an efficient method for integer factorization. This algorithm is widely used in cryptography for breaking down large numbers into their prime factors, which is a foundational aspect of many encryption systems. In addition to his work in factorization, Pollard has made contributions in various areas of mathematics, including algorithm design and computational complexity.

Lowell Schoenfeld is not a widely recognized or notable figure in public knowledge as of my last update. It is possible that he could be a private individual or a less-known person in a specific field.