Disproving Statement: No Natural Numbers Satisfy y^5 + 1 = (x^7-1)/(x-1)

Click For Summary
The discussion revolves around proving that there are no natural numbers x and y satisfying the equation y^5 + 1 = (x^7 - 1)/(x - 1). Participants simplify the equation and analyze the implications of prime factors and coprimality. They conclude that if x and c (derived from the equation) are both fifth powers, contradictions arise regarding their coprimality. Ultimately, they establish that no positive integer values for e exist that satisfy the equation, leading to the conclusion that the original statement is indeed disproven. The critical insight is that the factors involved cannot simultaneously satisfy the conditions required for both being fifth powers.
  • #31
Faiq said:
I don't think its necessary for both to be a fifth power consider this
y5 = (ab)5 = a5.b5
C = a5.b
x = b4
y5 = cx = (a5.b)(b4) = a5.b5 =(ab)5
But then C and x wouldn't be coprime they would have common factor b.
 
Physics news on Phys.org
  • #32
Oh yeah right thanks
 
  • #33
Btw you are right about case 1 and 3 we just omitted them because they kind of intuitively obvious that they lead to contradiction.
 
  • #34
Thank you very much for the help
 
  • Like
Likes Delta2

Similar threads

Where Do the Graphs of |3x-2| and 1/x Intersect and Diverge?
  • · Replies 4 ·
Replies
4
Views
2K
Finding the domain (and range) of a logarithmic function
  • · Replies 11 ·
Replies
11
Views
3K
Possible values an expression can take : ##\dfrac{x^2-x-6}{x-3}##
  • · Replies 7 ·
Replies
7
Views
1K
Express ##x^3+y^3## in terms of ##m## and ##n##
  • · Replies 17 ·
Replies
17
Views
3K
Finding the Parameters for the Curve y=asin(x-b)+c
  • · Replies 11 ·
Replies
11
Views
4K
Can Natural Numbers m and n Satisfy x + 1/m < y - 1/n?
  • · Replies 22 ·
Replies
22
Views
1K
Product of Missing Digits in a Number
  • · Replies 2 ·
Replies
2
Views
2K
What Steps Are Missing to Solve (2x+5)^(x+1)=729?
  • · Replies 11 ·
Replies
11
Views
3K
If ##x> 1## and ##x^2 <2##, prove ##x < y##, ##y^2<2##
  • · Replies 10 ·
Replies
10
Views
2K
Need help in manipulating rational absolute value inequalities
  • · Replies 11 ·
Replies
11
Views
2K