The statement in your question is not very precise. Obviously, ∏(√n) increments exactly when n=p2, where p is a prime, and for no other n. But I suppose that is not what you are looking for, and I am curious to know what your centered polygonal numbers have to do with this.JeremyEbert said:Is there a proof that ∏(√n) increments only when n is a centered polygonal number with a prime index?
Norwegian said:The statement in your question is not very precise. Obviously, ∏(√n) increments exactly when n=p2, where p is a prime, and for no other n. But I suppose that is not what you are looking for, and I am curious to know what your centered polygonal numbers have to do with this.
Norwegian said:Hi Jeremy, if we instead look at ∏(1/2 + √n), this function jumps exactly when n=(p-1/2)2=p2-p+1/4. If we require n to be an integer, the jumps will indeed happen when n=p2-p+1.
JeremyEbert said:Ah yes, the floor function is a bit redundant. Thanks for that.
This obviously relates then to the square root of a oblong (pronic) number + 1/4 having a half-integer value. The name "oblong" indicating they are analogous to polygonal numbers.
Anti-Crackpot said:I'm not sure what you mean by "analogous to polygonal numbers." Oblong numbers map 1:1 to triangular numbers (= 2 times a triangular number) and triangular numbers are one kind of polygonal number, but Polygonal numbers in general?
It's true that the polygonal numbers can be constructed from the counting numbers + multiples of triangular numbers, but that involves first differences between polygonal numbers of equal index (e.g. the 5th square is 25 and 25 + 10, the fourth triangular number, = 35, the 5th Pentagonal number), not polygonal numbers themselves.
- AC
The concept is based on the relationship between centered polygonal numbers and prime numbers. It shows that the product of all the square roots of the centered polygonal numbers with prime indices is equal to the increment between consecutive centered polygonal numbers.
This concept is important because it helps us understand the relationship between prime numbers and centered polygonal numbers, which are both important concepts in mathematics. It also provides a new way to approach and study these numbers.
Using prime indices in this concept is significant because it helps to narrow down the numbers being considered. Prime numbers have unique properties that make them fundamental in number theory, and by using their indices, we can focus on a specific set of centered polygonal numbers.
This concept can be applied in real-world situations in a variety of ways. For example, it can be used in cryptography to generate secure prime numbers, or in data encryption to create unique keys. It can also be applied in the study of patterns and sequences in nature, such as the arrangement of petals on a flower.
The potential future developments and implications of this concept are vast. It can lead to further discoveries and insights into the relationship between prime numbers and centered polygonal numbers. It may also have applications in other areas of mathematics and could potentially lead to new mathematical proofs and theorems.