Erdős Problem #470

Erdős Problem 470 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/470

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/470.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/470.tex.

1) FORMAL RESTATEMENT.

Let $\sigma(n)$ be the sum of positive divisors of $n$. Call $n$ weird if \[ \sigma(n)\ge 2n \] (i.e. $n$ is abundant or perfect) and $n$ is not pseudoperfect, i.e. $n$ is not the sum of any set of distinct proper divisors of $n$.

Call $n$ primitive weird if $n$ is weird and no proper divisor of $n$ is weird.

Questions:

1. Do there exist any odd weird numbers?
2. Are there infinitely many primitive weird numbers?

2) QUICK LITERATURE/CONTEXT CHECK.

The problem text states several known results: Benkoski--Erdős [BeEr74] proved weird numbers have positive density; Melfi [Me15] proved conditional infinitude of primitive weird numbers under a prime-gap hypothesis; Fang [Fa22] showed no odd weird numbers below $10^{21}$; Liddy--Riedl [LiRi18] proved an odd weird number must have at least 6 prime divisors. I will not assume any of these beyond what is explicitly stated.

3) ATTACK PLAN.

• Give a complete verification that $70$ is weird (and primitive weird) directly from its divisor set.
• Run an explicit search for weird numbers in a moderate range to sanity-check parity and primitiveness phenomena.
• Isolate crisp structural constraints that any odd weird number would have to satisfy (based only on definitions and verified computations).

4) WORK.

Lemma 470.1 ($70$ is weird and primitive weird). The integer $70$ is weird, and no proper divisor of $70$ is weird.

Proof. The positive divisors of $70$ are \[ 1,2,5,7,10,14,35,70. \] Hence \[ \sigma(70)=1+2+5+7+10+14+35+70=144\ge 140=2\cdot 70, \] so $70$ is abundant.

To check that $70$ is not pseudoperfect, consider its proper divisors \[ 1,2,5,7,10,14,35. \] If a subset of these sums to $70$, it must include $35$ (since the sum of the six proper divisors excluding $35$ is $1+2+5+7+10+14=39<70$). But if $35$ is included, the remaining needed sum is $35$. The largest possible sum using $\{1,2,5,7,10,14\}$ is $39$, and a direct check of combinations shows no subset sums to $35$: \[ 14+10+7+2+1=34,\quad 14+10+7+5=36, \] and all other subsets are smaller than $34$ or exceed $36$. Hence $35$ cannot be completed to $70$, so no subset of proper divisors sums to $70$. Therefore $70$ is weird.

Finally, every proper divisor of $70$ is one of $1,2,5,7,10,14,35$. Each has $\sigma(m)<2m$ (so is not abundant), hence cannot be weird. Thus $70$ is primitive weird. $\square$

Lemma 470.2 (computational parity check up to $200000$). There are no odd weird numbers $\le 200000$. Moreover, in this range there are $395$ weird numbers and $14$ primitive weird numbers.

Proof (explicit computation). For each $2\le n\le 200000$ I computed the list of proper divisors, checked abundance by comparing $\sum_{d\mid n,\ d<n} d$ with $n$, and checked pseudoperfectness by exact subset-sum DP over proper divisors (bitset method). A number was declared weird iff it was abundant and not pseudoperfect; it was declared primitive weird iff it was weird and no proper divisor was weird.

The results were:

count weird = 395
count primitive weird = 14
first 15 weird:
[70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792,
 10990, 11410, 11690, 12110, 12530]
first 14 primitive weird:
[70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356,
 73616, 83312, 91388, 113072]
odd weird found: []

So no odd weird numbers occur up to $200000$. $\square$

5) VERIFICATION.

• Lemma 470.1 is a complete hand-check using the explicit divisor list of $70$.
• Lemma 470.2 is an exhaustive check up to $200000$ using exact arithmetic and exact subset-sum DP, so it is a rigorous finite computation.
• The computational evidence is consistent with the statement in the problem text that no odd weird numbers exist below $10^{21}$ (though I did not verify that large bound).

6) FINAL. UNRESOLVED

(i) Strongest proved partial result. We have a complete proof that $70$ is (primitive) weird (Lemma 470.1), and an exhaustive search showing there are no odd weird numbers $\le 200000$ and listing the first several weird and primitive weird numbers (Lemma 470.2). The problem statement further records deep results (positive density, conditional infinitude of primitive weirds, and large-scale nonexistence results for odd weirds), but these are not proved here.

(ii) First gap (crisp). Either produce an explicit odd weird number (counterexample to "none exist"), or prove that no odd weird numbers exist. Independently, prove or disprove that there are infinitely many primitive weird numbers without additional hypotheses.

(iii) Top 3 next moves.

1. Try to find an unconditional infinite family of primitive weird numbers by proving a sufficient condition for weirdness in a parametrized family (e.g. numbers with controlled divisor structure), and then verifying primitiveness.
2. If aiming to rule out odd weirds, develop constraints for an odd abundant number to be non-pseudoperfect (subset-sum obstruction) and attempt to show these constraints cannot hold.
3. Extend computation: push the "no odd weird" search to higher bounds and analyze the prime-factor patterns of odd abundant numbers that fail/passed pseudoperfectness tests, to guide a theoretical obstruction.

(iv) Minimal counterexample structure. A minimal odd weird number (if it exists) would be an odd abundant number whose proper divisors fail every subset-sum representation of $n$; by the cited result in the problem text it would have at least 6 prime divisors. Any attempt to prove nonexistence must explain why every odd abundant integer with sufficiently many prime factors is nevertheless pseudoperfect.

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.

1) FORMAL RESTATEMENT.

Let $\sigma(n)$ be the sum of positive divisors of $n$. Call $n$ weird if \[ \sigma(n)\ge 2n \] (i.e. $n$ is abundant or perfect) and $n$ is not pseudoperfect, i.e. $n$ is not the sum of any set of distinct proper divisors of $n$.

Call $n$ primitive weird if $n$ is weird and no proper divisor of $n$ is weird.

Questions:

1. Do there exist any odd weird numbers?
2. Are there infinitely many primitive weird numbers?

2) QUICK LITERATURE/CONTEXT CHECK.

The problem text states several known results: Benkoski--Erdős [BeEr74] proved weird numbers have positive density; Melfi [Me15] proved conditional infinitude of primitive weird numbers under a prime-gap hypothesis; Fang [Fa22] showed no odd weird numbers below $10^{21}$; Liddy--Riedl [LiRi18] proved an odd weird number must have at least 6 prime divisors. I will not assume any of these beyond what is explicitly stated.

3) ATTACK PLAN.

• Give a complete verification that $70$ is weird (and primitive weird) directly from its divisor set.
• Run an explicit search for weird numbers in a moderate range to sanity-check parity and primitiveness phenomena.
• Isolate crisp structural constraints that any odd weird number would have to satisfy (based only on definitions and verified computations).

4) WORK.

Lemma 470.1 ($70$ is weird and primitive weird). The integer $70$ is weird, and no proper divisor of $70$ is weird.

Proof. The positive divisors of $70$ are \[ 1,2,5,7,10,14,35,70. \] Hence \[ \sigma(70)=1+2+5+7+10+14+35+70=144\ge 140=2\cdot 70, \] so $70$ is abundant.

To check that $70$ is not pseudoperfect, consider its proper divisors \[ 1,2,5,7,10,14,35. \] If a subset of these sums to $70$, it must include $35$ (since the sum of the six proper divisors excluding $35$ is $1+2+5+7+10+14=39<70$). But if $35$ is included, the remaining needed sum is $35$. The largest possible sum using $\{1,2,5,7,10,14\}$ is $39$, and a direct check of combinations shows no subset sums to $35$: \[ 14+10+7+2+1=34,\quad 14+10+7+5=36, \] and all other subsets are smaller than $34$ or exceed $36$. Hence $35$ cannot be completed to $70$, so no subset of proper divisors sums to $70$. Therefore $70$ is weird.

Finally, every proper divisor of $70$ is one of $1,2,5,7,10,14,35$. Each has $\sigma(m)<2m$ (so is not abundant), hence cannot be weird. Thus $70$ is primitive weird. $\square$

Lemma 470.2 (computational parity check up to $200000$). There are no odd weird numbers $\le 200000$. Moreover, in this range there are $395$ weird numbers and $14$ primitive weird numbers.

Proof (explicit computation). For each $2\le n\le 200000$ I computed the list of proper divisors, checked abundance by comparing $\sum_{d\mid n,\ d<n} d$ with $n$, and checked pseudoperfectness by exact subset-sum DP over proper divisors (bitset method). A number was declared weird iff it was abundant and not pseudoperfect; it was declared primitive weird iff it was weird and no proper divisor was weird.

The results were:

count weird = 395
count primitive weird = 14
first 15 weird:
[70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792,
 10990, 11410, 11690, 12110, 12530]
first 14 primitive weird:
[70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356,
 73616, 83312, 91388, 113072]
odd weird found: []

So no odd weird numbers occur up to $200000$. $\square$

5) VERIFICATION.

• Lemma 470.1 is a complete hand-check using the explicit divisor list of $70$.
• Lemma 470.2 is an exhaustive check up to $200000$ using exact arithmetic and exact subset-sum DP, so it is a rigorous finite computation.
• The computational evidence is consistent with the statement in the problem text that no odd weird numbers exist below $10^{21}$ (though I did not verify that large bound).

6) FINAL. UNRESOLVED

(i) Strongest proved partial result. We have a complete proof that $70$ is (primitive) weird (Lemma 470.1), and an exhaustive search showing there are no odd weird numbers $\le 200000$ and listing the first several weird and primitive weird numbers (Lemma 470.2). The problem statement further records deep results (positive density, conditional infinitude of primitive weirds, and large-scale nonexistence results for odd weirds), but these are not proved here.

(ii) First gap (crisp). Either produce an explicit odd weird number (counterexample to "none exist"), or prove that no odd weird numbers exist. Independently, prove or disprove that there are infinitely many primitive weird numbers without additional hypotheses.

(iii) Top 3 next moves.

1. Try to find an unconditional infinite family of primitive weird numbers by proving a sufficient condition for weirdness in a parametrized family (e.g. numbers with controlled divisor structure), and then verifying primitiveness.
2. If aiming to rule out odd weirds, develop constraints for an odd abundant number to be non-pseudoperfect (subset-sum obstruction) and attempt to show these constraints cannot hold.
3. Extend computation: push the "no odd weird" search to higher bounds and analyze the prime-factor patterns of odd abundant numbers that fail/passed pseudoperfectness tests, to guide a theoretical obstruction.

(iv) Minimal counterexample structure. A minimal odd weird number (if it exists) would be an odd abundant number whose proper divisors fail every subset-sum representation of $n$; by the cited result in the problem text it would have at least 6 prime divisors. Any attempt to prove nonexistence must explain why every odd abundant integer with sufficiently many prime factors is nevertheless pseudoperfect.