45^2 = 2025
Happy perfect square year, everyone. The previous one was 1936 and the next one will be 2116.
Happy perfect square year, everyone. The previous one was 1936 and the next one will be 2116.
1) is a square: 45²
2) is the product of two squares: 9² x 5²
3) is the sum of 3-squares: 40²+ 20²+5²
4) is the sum of cubes of all the single digits: 1³+2³+3³+4³+5³+6³+7³+8³+9³
6) sum of the first 45 odd numbers: 1+3+5+...+89
7) is a Harshad number: https://en.m.wikipedia.org/wiki/Harshad_number
3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers.
Edit: sorry, don't mean to be a pill.
You can prove it by induction; just expand (n(n+1)/2)² – (n(n-1)/2)², the result is n³.
Obviously all the formula will be equivalent to each other. They are, by construction, all restatements of each other.
Actually, property 5) trivially implies 1) but also 2), as `(1+2+...+n)² = n²(n+1)²/4` and either n or n+1 must be divisible by 2 hence one of the squares divisible by 4 hence it is a product of squares. But also property 4) as `(1+2+...+n)² = 1³+2³+...+n³` (easy to show by induction).
There is no such thing as a Harshad number, there is a _Harshad number in a given base_. All integers between zero and n are n-harshad numbers.
Which is a pity, because apparenty it means the `joy-giver`. I think human kind could use a joy giver year
[x**2 for x in range(32,100) if x**2 // 100 + x**2 % 100 == x]
[2025, 3025, 9801]At the ages of perfect squares is when we all cross or achieve significant milestones in our lives as children, students, (young)adults, spouses, parents, grandparents, senior citizens of society and so on.
This year being a perfect square, I wish that it will be as much or more special as it was for everyone at those ages.
If you multiply your ages… 36^2
And now, looking back 17 years later, I'd say he succeeded. It's the tarsnap founder.
1681: eight years later was glorious revolution with a bill of rights, marking individual freedoms
1764: ten years later, beginning of American Revolution and being free of monarchs
1849: ten years-ish later, start of US civil war; was the time of an attempt by the British to end slavery around the world
1936: ten years later, colonial empires were being dismantled, UN established to attempt global cooperation, US in the ascendancy with a seed of ties being established more by economics than military force, great economic upswing lifting people out of poverty (60% in poverty then, 10% now) while the global population blossoms
2035: Majority of the global population in middle class or better, triumph of individuals over technocrats, bureaucrats, and corporatists :)
(20+25)^(20/(2*5))