# This proves the claim.

• MHB
• Ackbach
In summary, "This proves the claim" is a statement that indicates strong evidence has been presented to support a particular claim in a scientific context. Proving a claim involves providing relevant and strong evidence through experiments, observations, or logical reasoning. While it may sound definitive, "This proves the claim" should be interpreted as meaning that the evidence strongly supports the claim, but it is not necessarily a definitive conclusion. It can be used in all scientific fields, but the terminology may vary. Finally, "This proves the claim" is not the final step in the scientific process, as further research and evidence-gathering may be necessary.
Ackbach
Gold Member
MHB
Here is this week's POTW:

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Prove that every nonzero coefficient of the Taylor series of
$\left(1 - x + x^2\right) e^x$
about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

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Congratulations to castor28 for his correct solution to this week's POTW, which is Problem A1 in the 2014 Putnam Archive. His solution follows:

[sp]
As the exponential series is absolutely convergent, we may multiply both series together.
If $a_n$ is the coefficient of $x^n$ in the resulting series, we have $a_0 = 1$, $a_1=0$, $a_2= \dfrac{1}{2}$. For $n>2$, we have:
\begin{align*}\displaystyle a_n &= \frac{1}{n!} - \frac{1}{(n-1)!} + \frac{1}{(n-2)!}\\ &= \frac{(n-1)^2}{n!}\\ &= \frac{n-1}{n(n-2)!} \end{align*}

and we must prove that, when this fraction is reduced to lowest terms, the numerator is $1$ or a prime number.

Writing $n = m+1$, we have:
$$\displaystyle a_n = \frac{m}{(m+1)(m-1)!}$$

We will write $\mathrm{N}(m)$ for the numerator of that reduced fraction. We must prove that this is $1$ or a prime.

The idea is to look at the prime divisors of $m$, and to estimate the number of occurrences of these in $(m-1)!$ to show that sufficient cancellation takes place.
We will use the well-known fact that, if $a$ and $b$ are integers with $a>1$ and $b> 0$, then $a^b > b$; furthermore, if $a>2$ or $b>2$, then we have the stronger relation $a^b > b+1$.

Case 1: $m = p$, $p$ prime

In this case, since $\gcd(p,(p-1)!) = \gcd(p, (p+1)) = 1$, no cancellation takes place and $\mathrm{N}(m)=p$.

Case 2: $m = p^k$, $p$ prime

The number of multiples of $p$ in the range $(1\dots m-1)$ is $p^{k-1}-1$. Note that these may include multiples of $p^2$, but we do not make use of this fact.
Using the fact mentioned above, we have:
\begin{align*}\displaystyle p^{k-1}&>k-1\\ p^{k-1}-1&>k-2\\ p^{k-1}-1 &\ge k-1 \end{align*}
This shows that $p^{k-1}\mid (m-1)!$, and $\mathrm{N}(m)=1 \text{ or } p$.
As mentioned above, if $p>2$ or $k>2$, we have actually $p^{k-1}>k$, and $\mathrm{N}(m)=1$.
Note that case 1 is actually a particular case of this case.

Case 3: $m = qp^k$, $p$ prime, $q>1$, $\gcd(p,q)=1$

The number of multiples of $p$ in the range $(1\dots m-1)$ is $qp^{k-1}-1 > p^{k-1}-1$. By comparison with case 2, we find that $p^k\mid (m-1)!$. As this holds for all primes that divide $m$, we conclude that $\mathrm{N}(m)=1$.

Rewriting all this in terms of $n$, the numerator of $a_n$ after reduction is:
• $1$ for $n=0\text{ or }2$
• $0$ for $n=1$
• $2$ for $n=5$
• $(n-1)$ when $(n-1)$ is prime
• $1$ otherwise
[/sp]

## 1. What does "This proves the claim" mean?

"This proves the claim" is a statement that suggests evidence or data has been presented to support a particular claim or hypothesis. It indicates that the evidence is strong enough to support the claim and that further investigation or research is not necessary.

## 2. How can one prove a claim?

Proving a claim involves providing evidence or data that supports the validity of the claim. This can be done through experiments, observations, or logical reasoning. The strength of the evidence and its relevance to the claim are important factors in proving a claim.

## 3. Is "This proves the claim" a definitive statement?

While "This proves the claim" may sound definitive, it is important to note that scientific claims are always subject to further investigation and could potentially be disproven in the future. Therefore, this statement should be interpreted as meaning that the evidence strongly supports the claim, but it is not necessarily a definitive conclusion.

## 4. Can "This proves the claim" be used in all scientific fields?

Yes, "This proves the claim" can be used in all scientific fields as long as the evidence presented is relevant and strong enough to support the claim. However, the terminology used may vary depending on the specific field of study.

## 5. Is "This proves the claim" the final step in the scientific process?

No, "This proves the claim" is not the final step in the scientific process. It is just one part of the process that involves making a claim, testing it, and presenting evidence to support it. The next step would be to continue researching and gathering evidence to further support or potentially disprove the claim.

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