The board / number theory / #1003
Erdős Problem #1003
OPENdatabase updated 2025-09-08
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Are there infinitely many solutions to \(\phi(n) = \phi(n+1)\), where \(\phi\) is the Euler totient function?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1003.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA001274
Original pageerdosproblems.com/1003
Attempts
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Exact radical-kernel bijection proved; all solutions with odd member <= 2^64-1 and omega(odd) <= 3 are exhaustively classified (15 total), but infinitude remains equivalent to constructing infinitely
PARTIAL