Rogers' Number Power Sum Problem, an extension of Fermat's last theorem.

I came up with an extension to Fermat’s last theorem.

I always thought that it was interesting that 

aⁿ + bⁿ = cⁿ where n was = 2, and 2 alone. 

3^2 + 4^2 = 5^2, is one of the solutions to Fermat's last theorem where n = 2, this is in fact Pythagoras's theorem for the diagonal for the hypotenuse in a right triangle. It has an infinite number of solutions. 

But then he took the exponent to 3 and up.  This always felt unbalanced to me. Like you need to add another term on the left each time you increment n.  My intuition feels like there has to be balance between the terms and the n value.

Roger’s Number Power Sum Problem:

For any positive integer n ≥ 2, consider the equation:

a₁ⁿ + a₂ⁿ + ... + aₙⁿ = zⁿ

Where a₁, a₂, ..., aₙ, and z are positive integers.

I tried it out with a simple example and it just worked. 

3^3 + 4^3 + 5^3 = 6^3
This does not fit the measurement of anything in a box or even a tesseract that I can figure out. 

The fact that this just worked the first  example simply by extending the same pattern is a very powerful sign to me that I am onto something.

That simple pattern breaks down at n=4 so far. My plan is to create a computer program to explore solutions. And to work on a general solution to this problem once I can see the patterns for solutions.  

And if this theorem only works for n=3 , then this is an amazing result too.  It would be very interesting to explore why it works for just one more example.  And if this is a general statement that shows there is a relationship between a power of n and n terms to the power of n, then that is also a very interesting result. 

I have a solution that proves this is not true for all positive numbers n>3, but the margin on this web page is too small to contain the answer.  (Just kidding. I don't plan on this being my last theorem.) 

And is 

aⁿ + bⁿ + cⁿ <> dⁿ for values of n bigger than 3? 

Is this a different proof than the proof that proved Fermat's last theorem? 

And is 

aⁿ + bⁿ + cⁿ + dⁿ <> eⁿ for values of n bigger than 4? 

Is this a different proof than the proof that proved the previous theorem, and that proved Fermat's last theorem? 

A Call for Constitutional Balance: Why Congress Must Act Against Presidential Immunity

The United States Constitution, revered as it should be, has been recently threatened by an unprecedented decision from our highest court. The notion that a sitting president is above criminal prosecution and accountability defies not only democratic principles but also the very fabric of American governance.

In this essay, I will argue against this ruling's misguided attempt to elevate the President into "a King" status, effectively rendering them immune to legal consequences for their actions while in office. Instead, Congress must take immediate action by affirming that no individual is above the law and ensuring our system remains grounded on equal justice under the Constitution.

Firstly, such a decision undermines the fundamental concept of checks and balances within government. The Founders deliberately crafted this framework to prevent any one branch from dominating others or exploiting its power for personal gain. By granting immunity to presidential actions, we risk creating an unchecked executive authority that can manipulate institutions at will (1). This flies in face of our nation's commitment to accountability and transparency.

Secondly, the notion of a "King-like" presidency would have far-reaching consequences on American politics as well as international relations. It could embolden authoritarian regimes worldwide by implying that even democratic leaders may operate with impunity when it suits their interests (2). Moreover, such an environment fosters corruption, cronyism, and abuse of power within our own government.

Thirdly, the Supreme Court's ruling will undermine public trust in institutions. As citizens learn about this decision, they will increasingly view elected officials as unaccountable to them – a sentiment that has already led to widespread disillusionment with politics (3). This erosion of faith can have devastating consequences for democratic stability and social cohesion.

Fourthly, it is crucial to recognize this decision as a gross overstep by our highest court in terms of its constitutional authority and jurisdiction. The Constitution explicitly grants Congress the power to impeach federal officials – including the President – which implies that they are not above the law (4). Furthermore, Article III ensures judicial review only within specific limits set forth in laws passed by Congress or treaties ratified with Senate approval; it does not grant courts carte blanche authority over political matters. By attempting to elevate any position beyond accountability through a constitutional amendment would require explicit Congressional authorization and ratification by three-fourths of the states – neither of which has occurred.

In conclusion, the recent Supreme Court ruling is an affront not only to our Constitution but also to fundamental principles of democracy. As Congress convenes in response to this alleged decision, it must take immediate action by affirming that no individual – including the President – stands above criminal prosecution or accountability for their actions while in office.

References:

(1) The Federalist Papers No. 51 (James Madison)

(2) "The Rise of Authoritarianism" - Foreign Affairs

(3) Pew Research Center: Public Trust in Government

(4) Article II, Section 2 & III, Clause 3