Mathematical conjecture & Proof of a conjecture 
1. A perfect square is a number that is the square of an integer.''
   conjecture & Proof of a conjecture 
   Let T(q): for Every natural number n the number is a perfect square of an integer if and only if its square root is a natural number'' T(q): number n is a perfect square if & only if its square root is a natural number.''
   & T(q): Every natural number n is a perfect square.''
   & T(q): number n is a perfect square'] & T(q): number is a perfect square.''
   conjecture for Every natural number n'] & Every natural number n.''
   & T(q): for Every natural number n'] & Every natural number n'] & T.'':'] & T.'': Every natural number n'].'': Every natural number n'] & T.'': Every natural number n'].'': Every natural number n T(q): number.'': Every natural number n'] & Every natural'] & T.'':'] & Every natural.'': Every natural.'':'].'': Every natural'] & T(q): number n.'': Every natural.'': Every natural'].'': Every natural'] & T.'':'].'': Every natural.'': Every natural.'':'] & T(q): number.'':'] & Every natural.'': Every natural.'': Every natural'].'':'] & T.'':'].'': Every natural.'':'].'':'] & Every natural'] & T(q): for Every natural.'':'] & Every natural.'':'] & T(q): number n'].'': Every natural.'': Every natural'] & Every natural'] & T(q): number n'].'': Every natural.'': Every natural'].'':'] & Every natural'] & T.'':'].'': Every natural.'':'].'': Every natural'].'':'].'': Every natural.'':'] & T.'':'].'': Every natural.'': Every natural'] & T(q): number n'].'': Every natural'] & Every natural.'':'] & T(q): number n'] & Every natural.'':'].'':'] & T.'':'] & T(q): number n.'': Every natural'] & T(q): number.'':'].'':'] & T(q): Every natural.'':'].'':'] & T.'':'] & Every natural.'':'] & T.'':'] & Every natural'].'': Every natural.'':']
It appears that the text you've provided is a jumbled and circular collection of statements related to the concept of a perfect square, but with excessive repetition, lack of clarity, and incorrect formatting. To address the original conjecture, let me break it down and provide a clear understanding of the concept.

**Original Conjecture:**

The statement to be proven is:

For every natural number n, the number is a perfect square if and only if its square root is a natural number.

**Corrected and Simplified Form:**

Simplified statement: For every natural number n, n is a perfect square if and only if ∛√n is a natural number.

**Why the simplification?**

The original conjecture was incorrect due to the circular reference of "Every natural number n". This was rewritten as n, which is a placeholder for any natural number. The square root of a number is √n, and for n to be a perfect square, √n must be a natural number.

**Proof of the Conjecture:**

Logical Proof:

(Only if) If n is a perfect square, then n = a^2 for some natural number a. This means that √n = a, which is a natural number since a is a natural number.

(Iff) If √n is a natural number, then n = (√n)^2 = a^2, which is a perfect square.

**Comment:**

This proof is a direct and simple argument based on the definition of perfect squares and natural numbers.