Pith Number

pith:ZMLJHR4S

pith:2026:ZMLJHR4SL53SJSP2LFN4724CII

not attested not anchored not stored refs resolved

Positive density for Sun's $2^k+m$ conjecture

Jinbo Yu, Songlin Han

Natural numbers n that can be written as n = k + m with 2^k + m prime have positive asymptotic density at least 0.0734.

arxiv:2605.15758 v1 · 2026-05-15 · math.NT

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\pithnumber{ZMLJHR4SL53SJSP2LFN4724CII}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We unconditionally prove that the natural numbers satisfying this property have a positive density. We compute this density to be at least 0.0734.

C2weakest assumption

The analytic or sieve estimates used to obtain the unconditional lower bound of 0.0734 are valid and produce a strictly positive quantity (as asserted in the unconditional claim).

C3one line summary

The set of n > 1 satisfying Sun's 2^k + m representation property has positive lower density at least 0.0734.

References

16 extracted · 16 resolved · 1 Pith anchors

[1] Kevin Broughan,Bounded gaps between primes: The epic breakthroughs of the early twenty-first century, Cambridge University Press, Cambridge, 2021. MR 4412547 2021

[2] Yong-Gao Chen and Xue-Gong Sun,On Romanoff’s constant, Journal of Number Theory106(2004), no. 2, 275–284 2004

[3] Roger Crocker,On the sum of a prime and of two powers of two, Pacific Journal of Mathematics36 (1971), no. 1, 103–107 1971

[4] Christian Elsholtz and Jan-Christoph Schlage-Puchta,On Romanov’s constant, Mathematische Zeitschrift288(2018), no. 3-4, 713–724 2018

[5] Paul Erdős,On integers of the form2k +pand some related problems, Summa Brasiliensis Mathe- maticae2(1950), no. 8, 113–123 1950

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:01:16.693091Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

cb1693c7925f7724c9fa595bcfeb82423dc3ecb5ec1a642051e663c3f68a565d

Aliases

arxiv: 2605.15758 · arxiv_version: 2605.15758v1 · doi: 10.48550/arxiv.2605.15758 · pith_short_12: ZMLJHR4SL53S · pith_short_16: ZMLJHR4SL53SJSP2 · pith_short_8: ZMLJHR4S

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ZMLJHR4SL53SJSP2LFN4724CII \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: cb1693c7925f7724c9fa595bcfeb82423dc3ecb5ec1a642051e663c3f68a565d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6cb171a026a822a9fc015b52c8dd4683213998c75a2f13a0e32c1a3e391b3961",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-15T09:18:26Z",
    "title_canon_sha256": "bd2144c4ad2c2dd4edae3ad8b936467f207459fb7535ddf5a0ff72a5426d2331"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15758",
    "kind": "arxiv",
    "version": 1
  }
}