Mathematicians Explore the Ulam Spiral and its Relatives

The Ulam spiral, a mathematical concept developed by Stanislaw Ulam in 1963, has been fascinating mathematicians for decades. The spiral is created by arranging natural numbers in a spiral pattern, with each number marked with a small circle if it is prime. Ulam observed that the spiral contains many straight lines, each of which contains a surprisingly large number of prime numbers.

The lines in the Ulam spiral correspond to sequences of the form 4n²+bn+c, where b and c are fixed integers and n=1,2,3,.... The coefficient b determines the orientation of the line: if b is odd, the line is vertical or horizontal, and if b is even, the line is diagonal with an inclination of 45° up or down.

Some of these sequences do not contain any prime numbers, such as 4n²+8n+3, which can always be written as a product (2n+1)×(2n+3). On the other hand, Ulam's observation suggests that for some values of the coefficients b and c, the sequence 4n²+bn+c contains many more prime numbers than expected.

In 1923, British mathematicians Godfrey H. Hardy and John E. Littlewood published a work that offers a possible explanation for this phenomenon. They conjectured that for every sequence 4n²+bn+c, there exists a constant A such that the number of prime numbers in the sequence less than a given large number N is approximately equal to A √N/logN. The larger the value of A, the more prime numbers exist in the corresponding line.

If prime numbers were distributed randomly among natural numbers, the constant A would be equal to 1/2 in all cases, regardless of the values of b and c. However, this is not what happens: in many cases, the value of A is much larger than 1/2, indicating that the number of prime numbers in the corresponding lines is much higher than expected.

A striking example is the sequence 4n²-2n+41, which corresponds to one of the most visible lines in the Ulam spiral. The value of A for this sequence is approximately 3.3, indicating that this line contains more than six times as many prime numbers as a typical line in the spiral.

Leonard Euler had observed a similar phenomenon in 1732, when he noticed that the polynomial q(m)= m²-m+41 seems to contain many prime numbers. Euler proved that q(m) is prime for all values of m from 1 to 40, but not for m=41. Interestingly, if we consider only the even values of m, the polynomial of Euler transforms into the sequence 4n²-2n+41, which is the same sequence we were discussing earlier.

This observation suggests that Euler's intuition was correct: the polynomial q(m) he proposed is indeed one of the sequences that contain many prime numbers. However, this is still a conjecture, as the work of Hardy and Littlewood is not yet proven.

Many other related problems remain open, including the question of whether the prime numbers of the form n²+1 form a finite or infinite set. We also do not know whether there exists any polynomial of degree 2 or higher that contains an infinite number of prime numbers. However, we do know that there is no polynomial that produces only prime numbers.

| **Sequence** | **b** | **c** | **A** | | --- | --- | --- | --- | | 4n²+8n+3 | 8 | 3 | 0.5 | | 4n²-2n+41 | -2 | 41 | 3.3 | | 4n²+bn+c | b | c | A |

Note: The table above is a summary of the sequences and their properties mentioned in the article. The values of b, c, and A are based on the text and are not necessarily exact values, but rather a representation of the relationships between the sequences and their properties.