The Fourth Compositional Iterate Modulo Eight

Abstract

Every coefficient above degree two in the fourth compositional iterate of A396798 is divisible by eight.

Let A denote D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries, the unique integer ordinary formal power series with zero constant coefficient satisfying A=X+I4(A)*I5(A). Write I0(F)=X and I(k+1)(F)=Ik(F) composed with F. The product is ordinary series multiplication, and the coefficients have no factorial scaling.

Theorem 1.1 (Hanna’s fourth conjecture).

$$\forall n:\mathbb{N},2<n⟹8\mid \mathrm{coeff}\left(n,\mathrm{iterate}\left(A,4\right)\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight.result (✓ std3). ∎

Resolves. Problems/oeis-a396798-fourth-iterate-mod-eight (proved) by D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A396798: compositional iterates modulo eight. URL: https://oeis.org/A396798.

Commentary.

Modulo four, source uniqueness identifies A with X/(1-X), so its second iterate equals X+2X^2. An exact integer quotient gives G=X+2X^2+4B for the second iterate modulo eight. Its zero constant coefficient makes substitution legitimate. The relations 4(G-X)=0 and 4((B composed with G)-B)=0, together with 2G^2=2X^2, yield G composed with G=X+4X^2. Thus every coefficient of the fourth iterate above degree two vanishes modulo eight.

References

• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries
• Truth anchor: D5/S1/Recurrence/Residue/IterateProductFourFiveFourthModEight.result
• Dependency: D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight