A factorization-dependent Landauer bound for multiplication, and why it cannot be exploited
Abstract
The Landauer principle is usually understood as a cost of computation: erasing information demands a minimum dissipation of heat. Here we ask whether this cost can itself carry information. We show that the minimal heat dissipated by a circuit that multiplies two integers and erases its inputs is set by the factorization structure of the product: products involving few primes are minimally irreversible and thus dissipate little heat, while highly composite products must dissipate more. Measuring the heat of a multiplier working at the Landauer limit would thus amount to a number-theoretic test. However, we argue that this signature is inaccessible in practice: a device approaching the Landauer limit would require computing all factorizations during its design, while a modular circuit with reusable components would not approach the Landauer bound. As an illustration, we estimate the heat generated by a multiplier in a positional base, showing that it destroys the factorization-specific signature and dissipates far more heat. The thermodynamic cost of multiplication therefore depends on factorization, but this dependence cannot be exploited by modular computing architectures.
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BibTeX
@misc{pau2026factorizationdependent,
title = {{A factorization-dependent Landauer bound for multiplication, and why it cannot be exploited}},
author = {Vilimelis Pau and Dominic Dall'Osto},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05743009v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A factorization-dependent Landauer bound for multiplication, and why it cannot be exploited
AU - Vilimelis Pau
AU - Dominic Dall'Osto
PY - 2026
DA - 2026/09/08
PB - HAL
UR - https://hal.science/hal-05743009v1
AB - The Landauer principle is usually understood as a cost of computation: erasing information demands a minimum dissipation of heat. Here we ask whether this cost can itself carry information. We show that the minimal heat dissipated by a circuit that multiplies two integers and erases its inputs is set by the factorization structure of the product: products involving few primes are minimally irreversible and thus dissipate little heat, while highly composite products must dissipate more. Measuring the heat of a multiplier working at the Landauer limit would thus amount to a number-theoretic test. However, we argue that this signature is inaccessible in practice: a device approaching the Landauer limit would require computing all factorizations during its design, while a modular circuit with reusable components would not approach the Landauer bound. As an illustration, we estimate the heat generated by a multiplier in a positional base, showing that it destroys the factorization-specific signature and dissipates far more heat. The thermodynamic cost of multiplication therefore depends on factorization, but this dependence cannot be exploited by modular computing architectures.
ER -