Uncovering the Mystery of 5 & 6: A Puzzle

[Surreal Ike]

What relationship do 5 and 6 have that make them unlike any other pair of distinct positive integers under 1000? (1000 was as far as I tested.)

WARNING: Don't read the rest of this thread if you want to solve the puzzle for yourself. It's not that hard--my dad got something rather close to the answer pretty quickly, and he's a computer programmer, not a mathematician. You can PM me if you want a hint.

 

[al-mahed]

is the only semi-prime number between twin prime numbers under 1000?

 

[Surreal Ike]

Please don't put any more answers in this thread.

Since you answer was wrong, I may as well respond to it. Both 4 and 6 are semiprime numbers that have prime numbers on both sides. Besides, you were supposed to find a relationship between the two numbers.

 

[al-mahed]

you're right, there are one more case... but seems to be the only 2 cases... (edited because I put some wrong statements)

if you show that thoses cases are not the only cases, then you'll be famous, because this could be a proof of the twin primes conjecture

ps: why we cannot post here?

 

[Surreal Ike]

I don't want anyone to post the answer here so that people can work out the answer for themselves.

But on second thought, maybe I will just insert a disclaimer in the first post.

 

[al-mahed]

well... ok... I think your puzzle is a little bit "generic", I mean, at least for me, as a first look, the puzzle can have multiple answers...

is some information missing?

 

[Surreal Ike]

No. I'm sorry it looks generic. Why don't you tell me all the answers you can think of, and I'll tell you whether you got the one I thought of.

Also, keep in mind that your answer has to express a unique relationship *between* the numbers. So just saying "6 is unique because x, and 5 is unique because y" is not going to cut it. The relationship I am talking about does not exist between any other pair of distinct numbers under 1000.

 

[Pere Callahan]

What about this:

The two positive integers x, y we are looking for satisfy

x^2-y^2 = (x-y)(x+y) = 11.

x=6, y=5 is clearly the unique solution.

There is no need for the restriction that the integers be less than 1000.

 

[Hurkyl]

What relationship do 5 and 6 have that make them unlike any other pair of distinct positive integers under 1000? (1000 was as far as I tested.)

That is the only (unordered) pair of distinct integers lying between 4 and 7.

 

[belliott4488]

Umm ... they're the only two sequential integers with the ratio 5/6?

 

[Pere Callahan]

The only pair of integers that equals (5,6) ...

 

[CRGreathouse]

Let W be the set of Wilson primes, and let $S=W\cup\{n:2n+1\in W\}$.

Then (5, 6) is the only pair of consecutive elements in S up to 500 million.

 

[CRGreathouse]

is the only semi-prime number between twin prime numbers under 1000?

4 is feeling pretty sad right about now. (4 and 6 are also kind of lonely: no company under 100 million.)

 

[Surreal Ike]

Guys, thanks for all the responses. The precondition can be stated in terms of all variables - no numbers necessary. (I guess I should have said this sooner, but it seemed like a bit of a hint.) That means *no* coefficients, only multiplication and addition of variables. But the answer can also be stated in plain english in a way that an intelligent middle schooler could understand. So you shouldn't try to figure it out using variables. Just think about what there could be, and keep coefficients and other numbers out of your answer.

Formally stated, the answer to the problem would go vaguely like this: 5 and 6 are the only numbers A and B such that [insert some equations involving constant variables A, B, C, D, E, and F involving NO COEFFICIENTS OR NONVARIABLE CONSTANTS].

There, I've given a good number of hints. Hopefully someone will PM me with the reply soon.

 

[Pere Callahan]

What do you mean by "constant variables" and "nonvariable constants"...?? And what don't you like about the sugestions presented so far?

 

[dodo]

A^2 - B^2 = A + B, which is the same Mr. Callahan said yesterday.

 

[Pere Callahan]

It is not the same because what you suggest only implies A-B = 1.

 

[rodigee]

Something cool.

The sum of the divisors of 5 plus the sum of the divisors of 6 divides the product of the divisors of 5 times the product of the divisors of 6.

$$(1 + 5 ) + (1+6+2+3)=18$$
$$(1 * 5 ) * (1*6*2*3)=180$$

 

[Ynaught?]

How about, 5 and 6 are the only consecutive numbers that are the sum and product (respectivley) of the same two prime numbers.

 

[CRGreathouse]

How about, 5 and 6 are the only consecutive numbers that are the sum and product (respectivley) of the same two prime numbers.

Good call. This is my favorite property so far on the thread. It works unordered, too.

 

[ramsey2879]

What relationship do 5 and 6 have that make them unlike any other pair of distinct positive integers under 1000? (1000 was as far as I tested.)

WARNING: Don't read the rest of this thread if you want to solve the puzzle for yourself. It's not that hard--my dad got something rather close to the answer pretty quickly, and he's a computer programmer, not a mathematician. You can PM me if you want a hint.

The only semiprime perfect number and the prime before it

 

[ramsey2879]

Something cool.

The sum of the divisors of 5 plus the sum of the divisors of 6 divides the product of the divisors of 5 times the product of the divisors of 6.

$$(1 + 5 ) + (1+6+2+3)=18$$
$$(1 * 5 ) * (1*6*2*3)=180$$

That property would be true for any prime number (n-1) where n is perfect but 5 and 6 are the only consecitive numbers where the smaller is prime and the larger perfect as 3 divides all other even perfect numbers less 1

 

[Pere Callahan]

But 8 and 7 have that property also

Any two consecutive integers have this property...

 

[Surreal Ike]

Ok, I'll just tell you guys my answer. Hopefully you won't be angry with me.

Stated in terms of integers C, D, E, and F, 5 and 6 are the only distinct integers such that

C + D = 5
C * D = 6
E + F = 6
E * F = 5

where c = 2, d = 3, e = 5, and f = 1.

In other words, 5 can be expressed as the sum of two of 6's divisors, and 6 can also be expressed as the sum of two of 5's divisors. Pretty simple, really.

The reason the problem said "distinct" is because 4 also has this property, in a way, since 2 + 2 = 4 and 2 * 2 = 4. Other than that, there is no other pair of numbers with this property under 1000.

I have no idea whether any of your other answers are intertwined with mine...

 

[ramsey2879]

Good call. This is my favorite property so far on the thread. It works unordered, too.

Ynaught's answer looks like the best answer to me also, i.e.

A,B are consecutive integers
P + Q = A
P * Q = B
P,Q are prime

 

[al-mahed]

x + y + 1 = xy ==> (y-1)x + y + 1 = 0 ==> (y+1)/(y-1) = x

call y - 1 = k ==> (k+2)/k = x ==> k(x-1) = 2 ==> x=3 and k=1 OR x=k=2 ==> x=3 and y=2 OR x=2 and y=3

showing (2,3) is the only pair such that x + y + 1 = xy

 

[al-mahed]

C + D = 5
C * D = 6
E + F = 6
E * F = 5

Hi surreal Ike,

This is superfluous since this property holds for every pair of consecutive positive integers

 

[al-mahed]

Ramsey, thanks for the warning, I really made a typo, correcting:

x + y + 1 = xy ==> (1-y)x + y + 1 = 0 ==> (y+1)/(y-1) = x

call y - 1 = k ==> (k+2)/k = x ==> k(x-1) = 2 ==> x=3 and k=1 OR x=k=2 ==> x=3 and y=2 OR x=2 and y=3

showing (2,3) is the only pair such that x + y + 1 = xy

 

[Surreal Ike]

al-mahed:

But not if C, D, E, and F all have to be integers. In that case, 5 and 6 are unique.

 

[al-mahed]

al-mahed:

But not if C, D, E, and F all have to be integers. In that case, 5 and 6 are unique.

what I meant is:

C + D = x
C * D = x + 1

is enought to make 5 and 6 unique as you state, and