Search Ideas
A smaller result from the same prompt (‘discover something nobody has discovered before’), again produced by Claude and not yet checked by a mathematician:
Claim: OEIS A281710 equals F(n+1) + 9 for all n ≥ 4, where F is the Fibonacci sequence (F(1) = F(2) = 1).
A281710 counts the n×3 arrays of 0s and 1s in which no cell differs from a strict majority of its horizontal, diagonal and antidiagonal neighbors, with the top-left cell fixed at 0. Its OEIS entry lists the recurrence a(n) = 2a(n−1) − a(n−3) for n > 6 and a generating function, both marked ‘Empirical’, meaning they were guessed from the data. The claim above implies both, so it would turn them into theorems.
Proof sketch:
- Whether a row is valid depends only on the rows directly above and below it. So the arrays can be counted with a transfer matrix whose states are pairs of adjacent rows. There are at most 64 such pairs. By the Cayley–Hamilton theorem, a(n) therefore satisfies a linear recurrence of order at most 64 for n ≥ 2.
- F(n+1) + 9 satisfies a linear recurrence of order 3.
- So the difference a(n) − F(n+1) − 9 satisfies a linear recurrence of order at most 67. A sequence like that is zero forever once it is zero for 67 terms in a row.
- Computing a(n) with the transfer matrix shows the difference is zero for every n from 4 to 300, which is 297 terms in a row.
Checks: a brute-force count over all arrays for n ≤ 6 agrees with the transfer matrix, and the computed terms match the terms listed in the OEIS entry.
Caveats: as of November 12, 2026, the OEIS entry shows no proof and doesn’t mention Fibonacci, but the wider literature was not searched, so an earlier proof may exist. Criticism is welcome.
having a good place to live and having good meals
The employer hires someone not for charity but because of the value they bring. That value isn’t suddenly increased by making hiring them more expensive.
having a good place to live and having good meals
Those goals are actually more attainable without minimum-wage laws. That’s because minimum wage laws, like most regulations, make it harder to create common preferences (and thus wealth). See #6133.
Those artists should get a job that pays more on its own, not get the government to force employers to pay them more. Otherwise one might as well argue that the government should force employers to pay employees a trillion dollars for one minute of work so they never have to worry about money again…
Nobody who works full time should be poor
Why should someone who works full time automatically not be poor? This idea assumes that working full time entitles a person to a certain income, but on what grounds? I could work full time doing something nobody wants to pay me for. Why would the hours I put in entitle me to pay above a government-set floor?
the minimum wage should be raised.
You can’t make someone richer by upping a number on a piece of paper. It’s a naive and mystical idea. The only legitimate way people get richer is by creating common preferences. That takes creativity, not the mechanical raising of numbers by fait/law. You can’t outlaw poverty. If you want people to get richer, reduce regulatory hurdles to make it easier for them to create common preferences.
Artists can focus on making art early on when they are starting out.
Imagine a singer in Nashville that is just starting out. If they are able to find a minimum wage that is decent enough for them not to think about having a good place to live and having good meals, they can instead focus on making music.
Since most wage earners are poor, the government should protect them, including by making sure their wages are enough to live on.
If I would like to work some job for $10/hour, but the state says I can’t accept the job for less than $11, at which rate it is not being offered, then the state is dooming me to be unemployed. It’s become my destroyer, not my protector.
must depend on the state.
The poor would have more voluntary charities to fall back if the welfare state hadn’t crowded them out. There used to be a flourishing network of charity organizations until the emergence of the mega welfare state in the 1960s.
the poor have nothing to fall back on and must depend on the state.
That doesn’t follow.
the minimum wage should be raised.
You can’t make someone richer by upping a number on a piece of paper. It’s a naive and mystical idea. The only legitimate way people get richer is by finding common preferences. That takes creativity, not the mechanical raising of numbers by fait/law. You can’t outlaw poverty. If you want people to get richer, reduce regulatory hurdles to make it easier for them to find common preferences.
The government should protect the poor. The rich can look after themselves, but the poor have nothing to fall back on and must depend on the state. Since most wage earners are poor, the government should protect them, including by making sure their wages are enough to live on.
Pope Leo XIII made this argument in his 1891 encyclical Rerum Novarum. He wrote that wage-earners “should be specially cared for and protected by the government” (§37), and that wages “ought not to be insufficient to support a frugal and well-behaved wage-earner” (§45).
Yet a full-time minimum-wage earner with two kids still falls below the poverty line, so the minimum wage should be raised.
That approach would at best treat symptoms, not the root cause. If prices are too high for people to afford basic things, the government should take a look in the mirror. It artificially raises prices through taxes, inflation, and bureaucracy.
cutting executive pay
This suggestion rests on the implicit idea that evil CEOs ‘steal’ value from their employees. That isn’t the case. Theft means taking someone’s property without consent. Employees instead voluntarily enter employment because they value the wages more than whatever else they have considered doing with their time instead.
Employment isn’t some abusive relationship where profit is ‘extracted’ from the employee. Under free-market conditions, the employer hires the employee, and the employee seeks employment from said employer, because both profit from the employment. A minimum wage, on the other hand, makes it harder for both to profit.
so the money to pay workers more is there
Even if #6118 were true, it doesn’t address the vast majority of businesses struggling to get by. The owners of mom-and-pop shops don’t make millions. Many businesses in affected industries (retail, restaurants) have razor-thin margins.
so the money to pay workers more is there
Even if this argument were true, it doesn’t address the vast majority of businesses struggling to get by. The owners of mom-and-pop shops don’t make millions. Many businesses in affected industries (retail, restaurants) have razor-thin margins.
This argument rests on the implicit idea that CEOs ‘steal’ value from their employees. That isn’t the case. Theft means taking someone’s property without consent. Employees instead voluntarily enter employment because they value the wages more than whatever else they have considered doing with their time instead.
Employers who can’t afford the higher wage have to let some workers go
Employers could pay for the higher wage by cutting executive pay instead of jobs. Many companies pay their CEOs millions, so the money to pay workers more is there, and laying workers off is a choice, not a necessity.
Raising the minimum wage would lift the incomes of millions of working families.
A higher minimum wage can lower incomes instead. Employers who can’t afford the higher wage have to let some workers go, and those workers then make less than before ($0), not more.
Raising the minimum wage would lift the incomes of millions of working families. For those families, it could mean the difference between groceries and the food bank, or between paying rent and being evicted.
Barack Obama made this argument in his 2013 State of the Union address, saying that raising the federal minimum wage to $9 an hour would “raise the incomes of millions of working families.”
Nobody who works full time should be poor. Yet a full-time minimum-wage earner with two kids still falls below the poverty line, so the minimum wage should be raised.
Barack Obama made this argument in his 2013 State of the Union address, declaring that “no one who works full-time should have to live in poverty.”
Nobody who works full time should be poor
Why shouldn’t someone who works full time be poor? This idea assumes that working full time entitles a person to a certain income, but on what grounds? I could work full time doing something nobody wants to pay me for. Why would the hours I put in entitle me to pay above a government-set floor?
Why shouldn’t someone who works full time be poor? This idea assumes that working full time entitles a person to a certain income, but on what grounds? I could work full time doing something nobody wants to pay me for. Why would the hours I put in entitle me to pay above a government-set floor?
A minimum wage reduces poverty. Nobody who works full time should be poor, yet a full-time minimum-wage earner with two kids still falls below the poverty line. Raising the minimum wage would lift the incomes of millions of working families.
Barack Obama made this argument in his 2013 State of the Union address, declaring that “no one who works full-time should have to live in poverty.”