My mind has gone fallow, and I can't quite understand factoring

In summary, the conversation revolves around understanding how to factor a number into two coprime numbers and what happens if the number of divisors is odd. The conversation also delves into the concept of consecutive pairs of residues in a sequence of numbers and how it relates to the pigeon hole principle. Despite several attempts to explain the concept, the individual seeking clarification still struggles to fully understand.
  • #36
matt grime said:
Have you considered checking a couple of counter examples, with say, p=2 or p=3, or b=0?

Why would p=2 or 3 be counterexamples?
 
Mathematics news on Phys.org
  • #37
it works iff a=/=b and a,b=/=0
 
  • #38
Sorry, but this is rubbbish: you have not even clearly stated a proposition. You cannot begin a sentence with if and only if. It makes no sense. And the best guess as to what you are conjecturing is trivially refuted with p=2 or p=3 without thinking very hard. When p=3 it reduces to a+b. So you're honestly attempting to claim that a+b is always 0 mod 3? Nonsense. Just think about it for more than one second and stop wasting people's time.
 
  • #39
matt grime said:
When p=3 it reduces to a+b. So you're honestly attempting to claim that a+b is always 0 mod 3? Nonsense. Just think about it for more than one second and stop wasting people's time.

When p=3, given that a,b<p, a=/=b, and a,b=/=0 then a+b=0mod3.
Is that incorrect?
 
  • #40
a=2 b=-1? That any good for you? State your question clearly. Are a and b supposed to be integers? residues mod p, what? Since you've used inequalities it can't be a statement about residues since they are not ordered (in any nice way).
 
Last edited:
  • #41
a and b must be positive integers. could you try to explain the proof of it?
 
  • #42
can anybody help me pleae?
 
  • #43
It's a geometric progression question, roger. You're supposed to recognise how to simplify

[tex]x^n+yx^{n-1}+\ldots y^n[/tex]

by thinking of geometric progressions. If you don't see why it is a geometric progression, try dividing by something.
 
  • #44
cheers matt.

But why must a and b be <p for it to work?
 
  • #45
a^3+b^5+c^7=d^11 iff a,b,c and d are natural numbers.

Is there a way to simplify this in order to find a,b,c and d?
and is there any significance in the powers being prime?
 
  • #46
roger said:
cheers matt.

But why must a and b be <p for it to work?

What makes you think they must be? Try finding some counter examples and try to spot a pattern. I already gave you many counter examples, and they all had the same theme to them.
 
  • #47
roger said:
a^3+b^5+c^7=d^11 iff a,b,c and d are natural numbers.

This makes no sense, again. You need to write things in sentences, and ones that make sense at that. We keep telling you this.

Are you asking to find the integer solutions to this?
 
  • #48
matt grime said:
This makes no sense, again. You need to write things in sentences, and ones that make sense at that. We keep telling you this.

Are you asking to find the integer solutions to this?

I'm asking firstly whether it can be simplified to find a,b,c, and d such that the equality is true, and secondly whether there is any significance in the powers being all prime?

But the 'counter examples you gave were based on any value of a and b whereas afterwards I stated that they must be positive integers.
 
  • #49
roger said:
a^3+b^5+c^7=d^11 iff a,b,c and d are natural numbers.

Is there a way to simplify this in order to find a,b,c and d?
and is there any significance in the powers being prime?

Your first statement is "a^3+ b^5+ c^7= d^11 iff a, b, c, and d are natural numbers". Do you understand that that asserts that a^3+ b^6+ c^7= d^11 for all natural numbers a, b, c, d? It certainly is NOT true since you can take a= b= c= d= 1 and get a false statement.

What you want to say is "Find natural numbers a, b, c, and d such that a^3+ b^5+ c^7= d^11 (or prove that no such numbers exist)".
 
  • #50
roger said:
But the 'counter examples you gave were based on any value of a and b whereas afterwards I stated that they must be positive integers.

I give up. You're on your own.
 
  • #51
can someone else help me please?
The counter examples were all false since a and b must be positive integers.
 
  • #52
What you want to say is "Find natural numbers a, b, c, and d such that a^3+ b^5+ c^7= d^11 (or prove that no such numbers exist)".

I understand, so can it be simplified? and is there any significance in the powers being all prime?
 
  • #53
p=2 or p=3 aren't counter examples since when p=2, it reduces to 1+1 so 2=0mod2 and when p=3, 2+1=0mod3.
 
  • #54
roger said:
can someone else help me please?
The counter examples were all false since a and b must be positive integers.
Seems I can't help myself. The point is that the counter examples all broke your initially unstated rules that 0<a<b<p. You asked why must a,b<p for this to work (which is a bad sentence by the way - use proper sentences). The answer is that they need not be, but that this condition, with the positivity condition will force it to work. Besides, you're just reducing everything mod p and so you only care about a,b mod p anyway.

The assertion will still be true for some values of a,b,p breaking that condition, but the triples where it fails all have something in common. Just look at them.

It will work for, say, a=-1,b=1 p=3. It won't work for a=0=b,p anything. It didn't work for a=-1,b=2,p=3, but it will for a=-1,b=2,p=5. Final hint: what is 2-(-1)?
 
Last edited:
  • #55
What if the conditions are that a and b must be positive integers in addition to being<p?
 
  • #56
wtf? I thought you'd proved it for 0<a<b<p.
 
  • #57
I have, but I just wanted to know why a and b had to be less than p. I couldn't see why it need to be.
 
  • #58
For hopefully the last time THEY DO NOT HAVE TO BE IN THE RANGE 0<a<b<p, but if they are not in that range they may or may not satisfy the condition. This is at least the 3rd time I've told you this.

It's a simple standard observation: A implies B, does not in mean that not(A) implies not(B).
 
  • #59
So my question is why is it that outside the range, they do not always work, but if they're inside the range, it will always work?
 
  • #60
But I've explained this to you as well. Look at the 'counter examples' outside the range. E.g. post 54. And don't think about posting another question on this until you've gone away and thought about why

a=-1,b=2,p=3 (and the fact that 2 - (-1)=3) is important. Look at your proof. Don't you divide by something at some point? Something that might be zero outside the specified range but isn't inside the specified range?
 
  • #61
The pattern is that a and p or b and p are coprime?

so how do I factorise this? a^3+ b^5+ c^7= d^11
 
  • #62
roger said:
The pattern is that a and p or b and p are coprime?

No. That is not at it at all. The counter example of a=-1, b=2 and p=3 disproves that assertion. Please, for the love of God, just think about it for one second.
 
  • #63
so can anybody explain to me the factorisation of a^3+ b^5+ c^7= d^11(or impossibilty thereof)?
 
  • #64
I think roger has exceeded his questions quota, and matt understandibly has lost his patience.
 
  • #65
how do you find the nth term and closed form sum of : (5/12)+(12/29)+(29/70)...?
 
  • #66
On the way to the definitive solution...
Obviously, numerator of the next fraction is the denominator of the previous fraction
Denominator of the next fraction equals numerator of this fraction plus sum of numerator and denominator of previous fraction
 
<h2>1. What does it mean for the mind to go fallow?</h2><p>When we say that the mind has gone fallow, we are referring to a state of mental inactivity or stagnation. It is a term often used to describe a lack of inspiration or creativity.</p><h2>2. How does a fallow mind affect understanding of factoring?</h2><p>A fallow mind can make it difficult to understand factoring because it can hinder our ability to think critically and problem solve. When our minds are not actively engaged, we may struggle to grasp new concepts or make connections between ideas.</p><h2>3. Can a fallow mind be a temporary state?</h2><p>Yes, a fallow mind can be a temporary state. It is often a result of mental fatigue or a lack of stimulation. Taking breaks, engaging in activities that promote creativity, and getting enough rest can help refresh the mind and overcome a state of mental fallowness.</p><h2>4. Are there any benefits to having a fallow mind?</h2><p>While a fallow mind can be frustrating, it can also have some benefits. It allows our brains to rest and recharge, which can improve overall mental well-being. It can also provide a break from constant mental stimulation, allowing us to come back to tasks with a fresh perspective.</p><h2>5. How can I overcome a fallow mind and improve my understanding of factoring?</h2><p>To overcome a fallow mind and improve understanding of factoring, it is important to engage in activities that stimulate the mind, such as reading, solving puzzles, or engaging in creative hobbies. It can also be helpful to take breaks and get enough rest to avoid mental fatigue. Additionally, seeking help from a tutor or mentor can provide additional support and guidance in understanding difficult concepts.</p>

1. What does it mean for the mind to go fallow?

When we say that the mind has gone fallow, we are referring to a state of mental inactivity or stagnation. It is a term often used to describe a lack of inspiration or creativity.

2. How does a fallow mind affect understanding of factoring?

A fallow mind can make it difficult to understand factoring because it can hinder our ability to think critically and problem solve. When our minds are not actively engaged, we may struggle to grasp new concepts or make connections between ideas.

3. Can a fallow mind be a temporary state?

Yes, a fallow mind can be a temporary state. It is often a result of mental fatigue or a lack of stimulation. Taking breaks, engaging in activities that promote creativity, and getting enough rest can help refresh the mind and overcome a state of mental fallowness.

4. Are there any benefits to having a fallow mind?

While a fallow mind can be frustrating, it can also have some benefits. It allows our brains to rest and recharge, which can improve overall mental well-being. It can also provide a break from constant mental stimulation, allowing us to come back to tasks with a fresh perspective.

5. How can I overcome a fallow mind and improve my understanding of factoring?

To overcome a fallow mind and improve understanding of factoring, it is important to engage in activities that stimulate the mind, such as reading, solving puzzles, or engaging in creative hobbies. It can also be helpful to take breaks and get enough rest to avoid mental fatigue. Additionally, seeking help from a tutor or mentor can provide additional support and guidance in understanding difficult concepts.

Similar threads

Replies
6
Views
774
  • General Math
Replies
12
Views
907
  • General Math
Replies
7
Views
1K
  • General Discussion
Replies
15
Views
1K
Replies
2
Views
1K
Replies
1
Views
932
Replies
19
Views
4K
Replies
24
Views
2K
  • General Math
Replies
12
Views
3K
  • General Math
Replies
13
Views
1K
Back
Top