The relationship of Counting number field & Prime numbers

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Counting number filed (1,2,3,....) results into consequences such as for example prime numbers. Prime numbers are located in the counting number filed (1,2,3,4,5,6,7,8,9,10,11,...). The bolded numbers are primes. So we can assume that the counting number field has a great relationship to PN that needs to study to know the deep nature of prime numbers.

So for you, what could be there relationship?

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Counting number filed (1,2,3,....) results into consequences such as for example prime numbers. Prime numbers are located in the counting number filed (1,2,3,4,5,6,7,8,9,10,11,...). The bolded numbers are primes. So we can assume that the counting number field has a great relationship to PN that needs to study to know the deep nature of prime numbers.

So for you, what could be there relationship?

Do you know what a mathematical field is?

The integers, both positive and negative, let alone the counting numbers do not satisfy the Field axiom for inverses for multiplication.

Edited by studiot
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Counting number filed field (1,2,3,....) results into consequences such as for example prime numbers. Prime numbers are located in the counting number filed field (1,2,3,4,5,6,7,8,9,10,11,...). The bolded numbers are primes. So we can assume that the counting number field has a great relationship to PN that needs to study to know the deep nature of prime numbers.

So for you, what could be there their relationship?

Primes are a subset of the set of natural numbers. (As studiot pointed out, 'field' has a specific meaning in mathematics beyond your apparent colloquial use.) There is no end to investigations into primes & their relations to natural numbers such as prime gaps, ending digits of primes [in whatever base], twin primes, etc. Pick your poison and drink deeply.

Edited by Acme
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Primes are a subset of the set of natural numbers. (As studiot pointed out, 'field' has a specific meaning in mathematics beyond your apparent colloquial use.) There is no end to investigations into primes & their relations to natural numbers such as prime gaps, ending digits of primes [in whatever base], twin primes, etc. Pick your poison and drink deeply.

If you are going to use a term in the Mathematics section, be expected to use terminology correctly. Otherwise, it's just random jargon use.

So for you, what could be there relationship?

If we knew that, the investigation into prime numbers wouldn't be such a huge problem.

Edited by Unity+
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Here is something more interesting than just criticising.

The formula

n2 + n + 41 where n is 0,1,2,3, 4 ..........

produces all prime numbers as far as n=39 and more prime numbers than any other quadratic formula as we run through the counting numbers.

This was due to Euler.

Edited by studiot
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Here is something more interesting than just criticising.

The formula

n2 + n + 41 where n is 0,1,2,3, 4 ..........

produces all prime numbers as far as n=39 and more prime numbers than any other quadratic formula as we run through the counting numbers.

This was due to Euler.

Mmmm...wouldn't n2 - n + 41 where n is 0,1,2,3, 4 ... produce the same number of primes?

Then there is the Ulam spiral & primes.

The primes of the form 4x2 − 2x + 41 with x = 0, 1, 2, ... have been highlighted in minority color. The prominent parallel line in the lower half of the figure corresponds to 4x2 + 2x + 41 or, equivalently, to negative values of x.

Hardy and Littlewood's Conjecture F

In their 1923 paper on the Goldbach Conjecture, Hardy and Littlewood stated a series of conjectures, one of which, if true, would explain some of the striking features of the Ulam spiral. This conjecture, which Hardy and Littlewood called "Conjecture F", is a special case of the BatemanHorn conjecture and asserts an asymptotic formula for the number of primes of the form ax2 + bx + c. Rays emanating from the central region of the Ulam spiral making angles of 45° with the horizontal and vertical correspond to numbers of the form 4x2 + bx + c with b even; horizontal and vertical rays correspond to numbers of the same form with b odd. Conjecture F provides a formula that can be used to estimate the density of primes along such rays. It implies that there will be considerable variability in the density along different rays. In particular, the density is highly sensitive to the discriminant of the polynomial, b2 − 16c.

...

An unusually rich polynomial is 4x2 − 2x + 41 which forms a visible line in the Ulam spiral. The constant A for this polynomial is approximately 6.6, meaning that the numbers it generates are almost seven times as likely to be prime as random numbers of comparable size, according to the conjecture. This particular polynomial is related to Euler's prime-generating polynomial x2 − x + 41 by replacing x with 2x, or equivalently, by restricting x to the even numbers. ...

Note: This source cites Euler's polynomial as the form I suggested as equivalent to studiot's. Are they equivalent in terms of the number of primes they produce?

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Mmmm...wouldn't n2 - n + 41 where n is 0,1,2,3, 4 ... produce the same number of primes?

Then there is the Ulam spiral & primes.

Note: This source cites Euler's polynomial as the form I suggested as equivalent to studiot's. Are they equivalent in terms of the number of primes they produce?

From a quick look I see Studiots produce till n=40 where it fails (1681 = 41^2 = 40^2+41+40) and yours till n=41 (1681 = 41^2= 41^2 - 41+41)

But yours produces 41 twice at n=0 and n=1 - so they are equivalent. I am pretty sure a little algebra would prove same

[hl]

it does

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