Can anyone tell the arithmetic progression where there are 10 prime numbers serially?

Answer 1

If the N numbers:

p, p + d, p + 2d, ..., p + (N-1)d

are all prime, and if q is a prime, then either:

q divides d and all elements of the sequence have the same residue modulo q (which would be nonzero unless N=1 and p=q)

or

any q consecutive elements of the sequence make up a complete residue set modulo q, i.e., their remainders modulo q will be the set {0, 1, 2, ..., 1-1} not necessarily in that order.

then note that unless 3 divides d that the first three taken modulo 3 form a complete residue set (i.e., the remainders modulo 3 will be the set {0, 1, 2} in some order). Likewise the first five taken modulo five will have remainders modulo five of {0, 1, 2, 3, 4} modulo five. So searches for N consecutive primes in arithmetic progression usually take N to be a multiple of all of the primes less than or equal to N.

A search for

prime number records consecutive arithmetic progression

in Google,

http://www.google.com/search?q=prime+number+records+consecutive+arithmetic+progression

yields as its first link

Prime Arithmetic Progression -- from Wolfram MathWorld
http://mathworld.wolfram.com/PrimeArithmeticProgression.html

which contains some sample sequences (smaller p and larger N, smaller N and larger p) and says it was last updated Friday, May 12, 2006 -- sample quote from that page:

"
For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.
"

(note that 210 = 2 * 3 * 5 * 7 is the product of all primes less than or equal to ten).

Dan D

Answer 2

Dan D did a good job, but there are lots of primes between the 10 primes he listed. If by "serially" you meant 10 consecutive primes, then I think this is what you were looking for:

http://www.ltkz.demon.co.uk/ar2/10primes.htm

Footnote: even if I gave you what you were looking for, go ahead and give Dan the 10 points. He deserves them. Cheers.

Answer 3

I don't think there is an arithmetic progression for the prime numbers.
the series would be the numbers itself but there is no specific formula to find n th number.

Answer 4

You could look into the rich prime-producing polynomial races.
The equation y=x^2+x+41 produces 39 prime numbers for x equals zero through 40.

There are many other rich prime producing polynomials that are interesting.

Answer 5

There is no formula, but you can use the Sieve of Erathosenes.
1. Make a list of nos. 1-100
2.Cross out 1.
3.Circle 2. Cross out all multiples of 2.
4.Go to next no. not crossed out, circle, and cross out all multiples.
5.Repeat Step 4 until all nos. are either circled or crossed.

Circled=prime
Crossed=Composite

Answer 6

Yes this really sound a convencing answer