**HP Prime: Fun with Primes**

**Prime Numbers**

A positive
integer is a prime number if there the only integers that can divide that
number evenly (without remainder) is 1 and itself.

Fun facts:

* 1 is not a prime number.

* 2 is the only even prime number.

* 5 is the only prime number that ends in
5.

* No prime number has a 0, 4, 6, or 8 as the last
digit.

**Sum of the Fist n Primes**

Let p be a
prime number. That is, p = {2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, …}

The sum of the first
prime numbers is: Σ p_k from k = 1 to n

HP Prime
Program SPRIMES: Sum of the First n
Primes

EXPORT SPRIMES(n)

BEGIN

// 2016-10-22 EWS

// Sum of the first
n primes

LOCAL t,p,k;

IF n≤1 THEN

RETURN 2;

ELSE

t:=2;

p:=2;

FOR k FROM 2 TO n DO

p:=CAS.nextprime(p);

t:=p+t;

END;

RETURN t;

END;

END;

**Sum of the First n Prime Reciprocals**

Σ 1/(p_k) from
k = 1 to n

HP Prime
Program ISPRIMES: Sum of first n Prime
Reciprocals

EXPORT ISPRIMES(n)

BEGIN

// 2016-10-22

// Sum of reciprocal
of primes

LOCAL t,p,k;

n:=IP(n);

IF n≤1 THEN

RETURN 2;

ELSE

t:=2¯¹;

p:=2;

FOR k FROM 2 TO n DO

p:=CAS.nextprime(p);

t:=p¯¹+t;

END;

RETURN t;

END;

END;

It does not
appear that there series of sums do not converge as n approaches ∞ (infinity).

ISPRIMES(25)
returns 1.80281720104

ISPRIMES(50)
returns 1.96702981491

ISPRIMES(100)
returns 2.10634212145

ISPRIMES(10000)
returns 2.70925824876

**Product of the First n Prime Reciprocals**

Π 1/(p_k) from
k = 1 to n

HP PRIME
Program IPPRIMES

EXPORT IPPRIMES(n)

BEGIN

// 2016-10-22

// Product of
reciprocal of primes

LOCAL t,p,k;

n:=IP(n);

IF n≤1 THEN

RETURN 2;

ELSE

t:=2¯¹;

p:=2;

FOR k FROM 2 TO n DO

p:=CAS.nextprime(p);

t:=p¯¹*t;

END;

RETURN t;

END;

END;

Unlike
ISPRIMES, IPPRIMES approaches 0 as n approaches ∞.

IPPRIMES(25)
returns 4.33732605429E-37

IPPRIMES(50)
returns 5.24156625851E-92

IPPRIMES(100)
returns 2.12227225409E-220

IPPRIMES(10000)
returns 0

This blog is
property of Edward Shore, 2016.

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