Abstract: The present invention is directed to a compound having antibacterial activity of the formula ##STR1## wherein R.sup.1 represents amino, protected amino, hydroxy or protected hydroxy;R.sup.4 represents heterocyclic thio wherein heterocyclic is a 5 to 6 membered hetero ring containing 1 to 4 hetero atoms from the group of oxygen, sulfur and nitrogen; andR.sup.5 is hydroxy or protected hydroxy, or a pharmaceutically acceptable salt thereof.

Abstract: For division by a Mersenne number (2.sup.n -1), a divided P is expressed as: ##EQU1## A first summing unit sums up the coefficients a.sub.i 's to provide a sum of a carry multiplied by 2.sup.n and a sum portion less than 2.sup.n. The carry is added to the sum portion to provide a similar sum. When the sum portion of such a sum becomes less than 2.sup.n, the sum portion provides the remainder if it is less than the Mersenne number. If not, the remainder is rendered zero with an additional carry produced. A second summing unit calculates ##STR1## for i=1 to (k-2). Each summation is given by a carry multiplied by 2.sup.n plus a sum portion with the carry from the first summing unit and the additional carry added to the summation for i=1 and with the carry from the summation i=i-1 added to the summation for i=i. The sum portions calculated by the second summing unit with successive addition of the carries provide zeroth through (k-2)-th coefficients to (2.sup.n).sup.O through (2.sup.n).sup.

Abstract: Novel cephem compounds of the formula; ##STR1## wherein R.sup.1 represents amino or hydroxyl group which may be protected, R.sup.2 represents amino or hydroxyl group or a group convertible into these groups, R.sup.3 represents hydrogen or methoxy group or a group convertible into methoxy group, R.sup.4 represents hydrogen or a residue of a nucleophilic compound and R.sup.8 represents hydrogen or a halogen, or a pharmaceutically acceptable salt or ester thereof, have strong antibiotic properties against a wide variety of microorganisms including gram-positive bacteria as well as gram-negative ones, especially by oral administration and can be used as therapeutic agent for various bacterial infections of animals including human beings.

Abstract: Novel cephem compounds of the formula; ##STR1## wherein R.sup.1 represents amino or hydroxyl group which may be protected, R.sup.2 represents amino or hydroxyl group or a group convertible into these groups, R.sup.3 represents hydrogen or methoxy group or a group convertible into methoxy group, R.sup.4 represents hydrogen or a residue of a nucleophilic compound and R.sup.8 represents hydrogen or a halogen, or a pharmaceutically acceptable salt or ester thereof, have strong antibiotic properties against a wide variety of microorganisms including gram-positive bacteria as well as gram-negative ones, especially by oral administration and can be used as therapeutic agent for various bacterial infections of animals including human beings.

Abstract: Novel cephem compounds of the formula; ##STR1## wherein R.sup.1 represents amino or hydroxyl group which may be protected, R.sup.2 represents amino or hydroxyl group or a group convertible into these groups, R.sup.3 represents hydrogen or methoxy group or a group convertible into methoxy group, R.sup.4 represents hydrogen or a residue of a nucleophilic compound and R.sup.8 represents hydrogen or a halogen, or a pharmaceutically acceptable salt or ester thereof, have strong antibiotic properties against a wide variety of microorganisms including gram-positive bacteria as well as gram-negative ones, especially by oral administration and can be used as therapeutic agent for various bacterial infections of animals including human beings.Also disclosed are novel thiazolylacetic acid compounds of the formula: ##STR2## wherein R.sup.1 represents amino, protected amino, hydroxyl or protected hydroxyl, R.sup.

Abstract: A compound of the formula: ##STR1## wherein R.sup.2 is hydrogen or a halogen and R.sup.3 stands for methyl, acetoxymethyl, carbamoyloxymethyl, an alkoxymethyl, an alkylthiomethyl, 2-carboxy-1-ethenyl, or a heterocyclic thiomethyl, or a pharmaceutically acceptable salt thereof, is found to have a broad antimicrobial spectrum and, in particular, effective against Gram-negative bacteria including Escherichia coli, Proteus vulgaris, Pseudomonas aeruginosa, Serratia marcescense as well as Gram-positive ones including antibiotic resistant strains. Thus, these compounds may be used for antimicrobial agents in therapeutical purposes.

Abstract: Novel cephem compounds of the formula; ##STR1## wherein R.sup.1 represents amino or hydroxyl group which may be protected, R.sup.2 represents amino or hydroxyl group or a group convertible into these groups, R.sup.3 represents hydrogen or methoxy group or a group convertible into methoxy group, R.sup.4 represents hydrogen or a residue of a nucleophilic compound and R.sup.8 represents hydrogen or a halogen, or a pharmaceutically acceptable salt or ester thereof, have strong antibiotic properties against a wide variety of microorganisms including gram-positive bacteria as well as gram-negative ones, especially by oral administration and can be used as therapeutic agent for various bacterial infections of animals including human beings.