- #1

kbannister

- 17

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- Homework Statement
- Find first 10 6-th cousins of the Giuga Numbers

- Relevant Equations
- kernel = sum(1./P)-1/X, a MATLAB expression where sum(1./P) = sum of the reciprocals

of the prime factors of a composite number having ONLY prime factors, e.g., 14, 21, or 55388487,

and X = the product of those prime factors, e.g., 3*7*11*347*691 = 55388487. In this latter

case, the RHS turns out to be = 4/7.

A Giuga Number is a positive integer x > 1 with prime factors p1,p2,p3,...pi

that satifies the relationship 1/p1 + 1/p2 + 1/p3 +...+ 1/pi - 1/x = k, where k is a

positive integer, k=1 in this case.

The first few Giuga numbers are 30, 858, 1772, and 66,198. For example, for x = 30 the prime factors are 2, 3, 5, so that we have

1/2 + 1/3 + 1/5 - 1/30 = 31/30 - 1/30 = 1 = k.

The problem now is to find "Cousins" of the Giuga numbers, namely the

first 10 unique combinations of primes, and their reciprocals, such

that a modified RHS, namely, k/(n+1), where n = 6, and 1 <= k an integer <= n, is

satisfied? For example, RHS values of 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 are OK.

By brute force looping in MATLAB, I have found the first 8 such numbers,

but #9 and #10 so far have eluded me. Also, 7 always appears as a factor. I believe the higher numbers might be very large - similar in behavior to the higher Giuga numbers.

I thought one way to speed up finding them is to simply use different "strides" in my

MATLAB loop, e.g., instead of 1, use 2*7, 3*7, 2*3*7, etc. I don't know, perhaps

there are cleverer ways based on more advanced knowledge about prime

number theory knowledge than I possess. I am an engineer, not a mathematician,

so prime number theory is not part of my kitbag of tools.

More about on Giuga numbers may be found in the on-line Encyclopedia

of Integer Sequences article: "Giuga numbers." The link is: https://oeis.org/A007850

that satifies the relationship 1/p1 + 1/p2 + 1/p3 +...+ 1/pi - 1/x = k, where k is a

positive integer, k=1 in this case.

The first few Giuga numbers are 30, 858, 1772, and 66,198. For example, for x = 30 the prime factors are 2, 3, 5, so that we have

1/2 + 1/3 + 1/5 - 1/30 = 31/30 - 1/30 = 1 = k.

The problem now is to find "Cousins" of the Giuga numbers, namely the

first 10 unique combinations of primes, and their reciprocals, such

that a modified RHS, namely, k/(n+1), where n = 6, and 1 <= k an integer <= n, is

satisfied? For example, RHS values of 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 are OK.

By brute force looping in MATLAB, I have found the first 8 such numbers,

but #9 and #10 so far have eluded me. Also, 7 always appears as a factor. I believe the higher numbers might be very large - similar in behavior to the higher Giuga numbers.

I thought one way to speed up finding them is to simply use different "strides" in my

MATLAB loop, e.g., instead of 1, use 2*7, 3*7, 2*3*7, etc. I don't know, perhaps

there are cleverer ways based on more advanced knowledge about prime

number theory knowledge than I possess. I am an engineer, not a mathematician,

so prime number theory is not part of my kitbag of tools.

More about on Giuga numbers may be found in the on-line Encyclopedia

of Integer Sequences article: "Giuga numbers." The link is: https://oeis.org/A007850

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