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.