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Probabilistic computing has been introduced to operate functional networks using a probabilistic bit (p-bit), generating 0 or 1 probabilistically from its electrical input. In contrast to quantum computers, probabilistic computing enables the operation of adiabatic algorithms even at room temperature, and is expected to broaden computational abilities in non-deterministic polynomial searching and learning problems. However, previous developments of probabilistic machines have focused on emulating the operation of quantum computers similarly, implementing every p-bit with large weight-sum matrix multiplication blocks or requiring tens of times more p-bits than semiprime bits. Furthermore, previous probabilistic machines adopted the graph model of quantum computers for updating the hardware connections, which further increased the number of sampling operations. Here we introduce a digitally accelerated prime factorization machine with a virtually connected Boltzmann machine and probabilistic annealing method, designed to reduce the complexity and number of sampling operations to below those of previous probabilistic factorization machines. In 10-bit to 64-bit factorizations were performed to assess the effectiveness of the machine, and the machine offers 1.2 X 10^8 times improvement in the number of sampling operations compared with previous factorization machines, with a 22-fold smaller hardware resource. This work shows that probabilistic machines can be implemented in a cost-effective manner using a field-programmable gate array, and hence we suggest that probabilistic computers can be employed for solving various large NP searching problems in the near future.