When a radio pulsar brakes down due to magnetodipole emission,its gravitational mass decreases accordingly. If the pulsar is hosted in a binary system, this mass loss will increase the orbital period of the system. We show that this relativistic effect can be indeed observable if the neutron star is fast and magnetized enough and that, if observed, it will help to put tight constraints to the equation of state of ultradense matter. Moreover, in Low Mass X-ray Binaries that evolve towards short

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... eriods, the neutron star lights up as a radio pulsar during the "period gap". As the effect we consider contrasts the orbital period decay, the system spends a longer time in this phase. As a consequence, the neutron star can survive this phase only if it is non-supramassive. Since in such bianries ∼ 0.8M can be accreted onto the neutron star, short period (P ≤ 2 h) millisecond Xray pulsars like SAX J1808.4-3658 can be formed only if either a large part of the accreting matter has been ejected from the system, or the equation of state of ultradense matter is very stiff.