Can you prove that x^3-8 is prime if x and x^2+8 are also primes?

  • Thread starter Nexus[Free-DC]
  • Start date
In summary, the conversation discusses a question about proving a statement involving primes and high school math. The conversation includes an initial solution from cookiemonster and a second solution from Zurtex. The conversation also touches on the topic of proving that x and x^2+8 cannot both be prime numbers.
  • #1
Nexus[Free-DC]
37
0
As a little diversion I thought i'd post this question which I call a forehead-slapper because that's what you'll likely do when you see the answer. You won't need more than high school maths to solve it.


Show that if x and [tex]x^2+8[/tex] are primes then so is [tex]x^3-8[/tex]
 
Mathematics news on Phys.org
  • #2
x = 3?

cookiemonster
 
  • #3
Well [itex]x^3 - 8 = (x - 2)(x^2 + 2x + 4) [/itex] meaning that this can only be a prime when [itex]x = 3[/itex].

Edit: I think I have proved the rest of it I'll let others have a go.
 
Last edited:
  • #4
There's a rest of it?

cookiemonster
 
  • #5
Given the Zurtex factorization, it seems pretty obvious that for x=4, 5, 6, ..., x^3-8 has to be composite, so I'm with Cookiemonster.
 
  • #6
Zurtex is exactly right. That was exactly the solution I had in mind. There is another one though I just realized, that doesn't involve having to factor a cubic.

Note that any prime p except 3 is equal to 3k+1 or 3k-1 for some integer k. Then p^2 + 8 = 9k^2 ± 6k + 9, which is divisible by 3. 3^2+8=17, which is prime and also 3^3-8=19 is prime.
 
  • #7
Nexus[Free-DC] said:
Zurtex is exactly right. That was exactly the solution I had in mind. There is another one though I just realized, that doesn't involve having to factor a cubic.

Note that any prime p except 3 is equal to K+1 or K-1 for some integer k. Then p^2 + 8 = K^2 ± K + 9, which is divisible by 3. 3^2+8=17, which is prime and also 3^3-8=19 is prime.
Yes, the rest of it was obviously to prove that [itex]x[/itex] and [itex]x^2 + 8[/itex] could never both be prime. My proof was a little bit more complex that as it is early in the morning and I can't think simple maths yet :rolleyes:
 
  • #8
Nexus[Free-DC] said:
Show that if x and [tex]x^2+8[/tex] are primes then so is [tex]x^3-8[/tex]

The point of cookiemonster's post was that x= 3 is a prime number and that x2[/sup+8= 9+8= 17 is a prime number but x3= 27-8= 21= 3*7 is NOT.

You can't prove your statement: it's not true.

What Zurtex showed with "[itex]x^3 - 8 = (x - 2)(x^2 + 2x + 4) [/itex]" was that x3- 8 cannot be prime unless x= 3. That is essentially a converse of your original statement.
 
  • #9
27-8 = 19 <filler space>
 

FAQ: Can you prove that x^3-8 is prime if x and x^2+8 are also primes?

What is "Forehead slapper #1"?

"Forehead slapper #1" is a term used to describe a common scientific phenomenon where a person realizes an obvious solution or answer to a problem that they had previously overlooked.

Why is it called a "forehead slapper"?

It is called a "forehead slapper" because the realization of the obvious solution or answer can be so clear and simple that the person may slap their forehead in frustration for not realizing it sooner.

Is "forehead slapper #1" a real scientific term?

No, "forehead slapper #1" is not a formal scientific term. It is a colloquial term used by scientists and researchers to describe a common experience.

What causes "forehead slapper #1" moments?

There are several factors that can contribute to a "forehead slapper #1" moment, including a lack of information, cognitive biases, and previous assumptions or beliefs. These moments can also be triggered by new perspectives or insights.

How can "forehead slapper #1" moments be beneficial in science?

"Forehead slapper #1" moments can help scientists and researchers to think critically and question their assumptions. They can also lead to new discoveries and breakthroughs in scientific understanding.

Similar threads

MHB How can you solve $5^{x^2+8}=125^{2x}$ for x?
Replies
8
Views
985
MHB Is $p^3+4$ Prime if Both $p$ and $p^2+8$ are Prime Numbers?
Replies
4
Views
2K
Solving for x in 0 = -1/8 + 3/8 x / (10^2 +x^2)^0.5
Replies
10
Views
2K
MHB Are there infinitely many primes that satisfy $p=3$ mod4 and divide $x^2+2$?
Replies
5
Views
2K
MHB Can you prove the following two difficult trigonometric identities?
Replies
3
Views
1K
Show N has Prime Numbers
Replies
4
Views
2K
MHB Year 7 & 8 mathematics puzzle solve (1−1/2)(1−1/3)....(1−1/2017)=x
Replies
5
Views
2K
MHB As for the question,How can I simplify (8x^1/2)/(x^2/3)]^1/3?
Replies
5
Views
2K
MHB Can You Prove $a > \sqrt[9]{8}$ is a Root of a Polynomial with $1 < a < 2$?
Replies
4
Views
2K
MHB Solve Equation: (7x+1)^(1/3)+(-x^2+x+8)^(1/3)+(x^2-8x-1)^(1/3)=2
Replies
6
Views
2K

Hot Threads

Recent Insights

Back
Top