In November last year, Chilean mathematician Hector Pasteu successfully solved a complex problem that had challenged him for over a decade. Surprisingly, the breakthrough came to him while procrastinating on preparing an exam for his students, as he pondered his favorite numerical sequence: 2, 5, 10, 17, 26, and so on. This sequence follows the pattern of N² + 1, where n is a whole number.
The sequence N² + 1 has been a subject of interest in number theory for a long time, as it combines two fundamental aspects of mathematics: addition and multiplication. These operations in number theory often lead to profound and unexpectedly intricate problems. For instance, one of the most significant unresolved mathematical tasks is the hypothesis that every even number greater than 2 can be expressed as the sum of two prime numbers. Another important question is the infinitude of twin prime pairs that differ by 2, like 11 and 13.
Despite its seemingly simple nature, the N² + 1 sequence conceals many mysteries. One of the central questions that has intrigued mathematicians for over a century is whether this sequence contains an infinite number of prime numbers. While the basic properties of the sequence have been understood for a long time, numerous inquiries remain unanswered, such as the rate at which prime numbers occur within this sequence.
Pasteu focused on proving that the numbers in the N² + 1 sequence always possess at least one large prime factor. This presented a formidable challenge due to the unpredictable behavior of prime numbers. His breakthrough involved applying novel methods to control the exponents in the prime factorization of numbers, resulting in significant progress.
By utilizing elliptic curves and modular forms, Pasteu developed a technique that demonstrated the largest prime factor of numbers in the N² + 1 sequence grows much faster than previously anticipated. This advancement not only contributed to the understanding of a specific numerical sequence but also opened up new possibilities for tackling other mathematical problems related to the interplay of addition and multiplication.
Pasteu’s work has garnered considerable attention in the scientific community. His research was published in the prestigious mathematical journal Inventiones Mathematicae, underscoring the significance of his discovery. Scientists have noted that this breakthrough could lead to further advancements in the exploration of numerical sequences and the resolution of other unresolved issues