Proving the Last Gauss Lemma for Real Numbers Using Induction

In summary, the formula [a,b,...,l,m]*[b,c,...,l] - [a,b,...,l]*[b,c,...,m] = (-1)^n is proven for real numbers a, b, c,..., with [a] = a, [a,b] = b*a+1, [a,b,c] = c*[a,b]+[a], [a,b,c,d] = d*[a,b,c]+[a,b], etc. The proof involves defining variables and using induction to show that the formula holds for all n. Thanks to the moderators for their help.
  • #1
Ben2
37
9
Modified from Disquisitiones Arithmeticae, p. 10: Let "*" indicate multiplication and "^" indicate "to the power." For real numbers a, b, c,..., let [a] = a, [a,b] = b*a+1, [a,b,c] = c*[a,b]+[a], [a,b,c,d] = d*[a,b,c]+[a,b], etc. Prove that [a,b,...,l,m]*[b,c,...,l] - [a,b,...,l]*[b,c,...,m] = (-1)^n, where n is the number of elements in the set {a,b,...,l,m}.
Thanks to the moderators for their help, and I won't post this again.
 
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  • #2
Ben2 said:
Modified from Disquisitiones Arithmeticae, p. 10: Let "*" indicate multiplication and "^" indicate "to the power." For real numbers a, b, c,..., let [a] = a, [a,b] = b*a+1, [a,b,c] = c*[a,b]+[a], [a,b,c,d] = d*[a,b,c]+[a,b], etc. Prove that [a,b,...,l,m]*[b,c,...,l] - [a,b,...,l]*[b,c,...,m] = (-1)^n, where n is the number of elements in the set {a,b,...,l,m}.
Thanks to the moderators for their help, and I won't post this again.
In modern notation we have the following situation:
Given two numbers ##a,b## we define:
$$
a_{-1}=0\, , \,a_0=1\, , \,a_1=a\, , \,a_n=b_n\cdot a_{n-1}+a_{n-2} \text{ and } \\
b_1 = a\, , \,b_2=b\, , \,b_n=\dfrac{a_n-a_{n-2}}{a_{n-1}} \\
c_{-2}=1\, , \,c_{-1}=0\, , \,c_0=1\, , \,c_1=b\, , \,c_n=b_{n+1}\cdot c_{n-1}+c_{n-2}
$$
and the statement then reads
$$
a_n\cdot c_{n-2}-a_{n-1}\cdot c_{n-1} = (-1)^n\quad (n \geq 0)
$$
and the rest is induction where the actual value of ##b_n## isn't relevant anymore.
 
Last edited:

Related to Proving the Last Gauss Lemma for Real Numbers Using Induction

1. What is the purpose of "Last Gauss Lemma Section II"?

The purpose of "Last Gauss Lemma Section II" is to explore the concept of Gauss Lemma in mathematics and its applications in various fields such as number theory and algebraic geometry.

2. What is Gauss Lemma?

Gauss Lemma is a fundamental theorem in number theory that states if a polynomial with integer coefficients can be factored into two relatively prime polynomials over the rational numbers, then it can also be factored into two relatively prime polynomials over the integers.

3. What are the applications of Gauss Lemma?

Gauss Lemma has various applications in number theory, algebraic geometry, and commutative algebra. It is used to prove important theorems such as Eisenstein's criterion, Hensel's lemma, and the fundamental theorem of algebra. It also has applications in cryptography and coding theory.

4. How does "Last Gauss Lemma Section II" differ from the first section?

The first section of Gauss Lemma deals with the basic concepts and proofs, while the last section delves deeper into the applications and extensions of the lemma. It may also cover more advanced topics such as quadratic forms and quadratic reciprocity.

5. Are there any prerequisites for understanding "Last Gauss Lemma Section II"?

A basic understanding of number theory, algebra, and polynomial functions is recommended for understanding "Last Gauss Lemma Section II". Familiarity with concepts such as prime numbers, factorization, and polynomial division will also be helpful.

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