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Magnetic fields are observed everywhere in the present universe. Among them, the intergalactic magnetic field in voids is likely separated from astrophysics and therefore implies its cosmological origin. Many mechanisms for generating the primordial magnetic field in the early universe have been proposed, which may be tested by the observation of the intergalactic magnetic field. I will discuss how the scenarios of magnetogenesis are constrained in consistency with the BBN, by making use of the recently proposed conserved quantity, Hosking integral.