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

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SUMMARY

The forum discussion centers on 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 explore the implications of prime factorization, particularly focusing on the relationship between x and y. Key insights include the necessity for x and c (where c = x^5 + x^4 + x^3 + x^2 + x + 1) to be coprime and the conclusion that if both are fifth powers, contradictions arise, confirming that no such natural numbers exist.

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  • Familiarity with the Fundamental Theorem of Arithmetic.
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  • #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.
 
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  • #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
 
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